Search: a028488 -id:a028488
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A008836
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Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).
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+10
191
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1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1
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OFFSET
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1,1
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COMMENTS
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Coons and Borwein: "We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non-trivial completely multiplicative function from N to {-1,1}, the series sum_{n=1..infinity} f(n)z^n is transcendental over {Z}[z]; in particular, sum_{n=1..infinity} lambda(n)z^n is transcendental, where lambda is Liouville's function. The transcendence of sum_{n=1..infinity} mu(n)z^n is also proved." - Jonathan Vos Post, Jun 11 2008
Coons proves that a(n) is not k-automatic for any k > 2. - Jonathan Vos Post, Oct 22 2008
The Riemann hypothesis is equivalent to the statement that for every fixed epsilon > 0, lim_{n -> infinity} (a(1) + a(2) + ... + a(n))/n^(1/2 + epsilon) = 0 (Borwein et al., theorem 1.2). - Arkadiusz Wesolowski, Oct 08 2013
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 1-11.
H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
P. Ribenboim, Algebraic Numbers, p. 44.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 112.
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LINKS
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]
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FORMULA
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Dirichlet g.f.: zeta(2s)/zeta(s); Dirichlet inverse of A008966.
Sum_{ d divides n } lambda(d) = 1 if n is a square, otherwise 0.
Completely multiplicative with a(p) = -1, p prime.
Recurrence: a(1)=1, n > 1: a(n) = sign(1/2 - Sum_{d<n, d|n} a(d)). - Mats Granvik, Oct 11 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} mu(n/d)^2 * a(d). - Ilya Gutkovskiy, Mar 10 2021
a(n) = sin(tau(n^2)*Pi/2). (End)
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EXAMPLE
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a(4) = 1 because since bigomega(4) = 2 (the prime divisor 2 is counted twice), then (-1)^2 = 1.
a(5) = -1 because 5 is prime and therefore bigomega(5) = 1 and (-1)^1 = -1.
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MAPLE
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with(numtheory): A008836 := proc(n) local i, it, s; it := ifactors(n): s := (-1)^add(it[2][i][2], i=1..nops(it[2])): RETURN(s) end:
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MATHEMATICA
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Table[If[OddQ[PrimeOmega[n]], -1, 1], {n, 110}] (* Harvey P. Dale, Sep 10 2014 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, n=factor(n); (-1)^sum(i=1, matsize(n)[1], n[i, 2]))}; /* Michael Somos, Jan 01 2006 */
(Haskell)
(Python)
from sympy import factorint
def A008836(n): return -1 if sum(factorint(n).values()) % 2 else 1 # Chai Wah Wu, May 24 2022
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CROSSREFS
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Cf. A000005, A001222, A002053, A007421, A002819 (partial sums), A008683, A010052, A026424, A028260, A028488, A056912, A056913, A065043, A066829, A106400, A156552, A349905, A063647, A101455, A048691.
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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STATUS
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approved
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A002819
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Liouville's function L(n) = partial sums of A008836.
(Formerly M0042 N0012)
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+10
31
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0, 1, 0, -1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, -2, -3, -4, -3, -2, -3, -2, -1, 0, -1, -2, -3, -4, -5, -6, -5, -4, -3, -2, -3, -2, -1, 0, -1, -2, -3, -4, -5, -4, -5, -6, -5, -6, -5, -6, -7, -6, -5, -4, -3, -2, -3, -2, -3, -2, -3, -2, -1, -2, -3, -4, -3, -4, -5, -6, -7, -6, -7, -8, -7, -8, -9, -10, -9, -8, -9, -8, -7, -6
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OFFSET
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0,9
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COMMENTS
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Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo. George Polya conjectured 1919 that L(n) <= 0 for all n >= 2. The conjecture was generally deemed true for nearly 40 years, until 1958, when C. B. Haselgrove proved that L(n) > 0 for infinitely many n. In 1962, R. S. Lehman found that L(906180359) = 1 and in 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257. - Harri Ristiniemi (harri.ristiniemi(AT)nicf.), Jun 23 2001
Prime number theorem is equivalent to a(n)=o(n). - Benoit Cloitre, Feb 02 2003
In the Liouville function, every prime is assigned the value -1, so it may be expected that the values of a(n) are minimal (A360659) among all completely multiplicative sign functions. As it turns out, this is the case for n < 14 and n = 20. For any other n < 500 there exists a completely multiplicative sign function with a sum less than that of the Liouville function. Conjecture: A360659(n) < a(n) for n > 20. - Bartlomiej Pawlik, Mar 05 2023
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REFERENCES
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]
H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
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FORMULA
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MAPLE
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MATHEMATICA
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Accumulate[Join[{0}, LiouvilleLambda[Range[90]]]] (* Harvey P. Dale, Nov 08 2011 *)
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PROG
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(PARI) a(n)=sum(i=1, n, (-1)^bigomega(i))
(PARI) a(n)=my(v=vectorsmall(n, i, 1)); forprime(p=2, sqrtint(n), for(e=2, logint(n, p), forstep(i=p^e, n, p^e, v[i]*=-1))); forprime(p=2, n, forstep(i=p, n, p, v[i]*=-1)); sum(i=1, #v, v[i]) \\ Charles R Greathouse IV, Aug 20 2016
(Haskell)
a002819 n = a002819_list !! n
a002819_list = scanl (+) 0 a008836_list
(Python)
from functools import reduce
from operator import ixor
from sympy import factorint
def A002819(n): return sum(-1 if reduce(ixor, factorint(i).values(), 0)&1 else 1 for i in range(1, n+1)) # Chai Wah Wu, Dec 19 2022
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CROSSREFS
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KEYWORD
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nice,sign
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jul 09 2001
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STATUS
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approved
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A051470
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a(n) is least value of m for which the sum of Liouville's function from 1 to m is n.
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+10
12
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1, 906150258, 906150259, 906150260, 906150263, 906150264, 906150331, 906150334, 906150337, 906150338, 906150339, 906150358, 906150359, 906150362, 906150363, 906150368, 906150387, 906150388, 906150389, 906150406, 906150407
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OFFSET
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1,2
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COMMENTS
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It was once conjectured that the sum of Liouville's function was never > 0 except for the first term.
It follows from Theorem 2 in Borwein-Ferguson-Mossinghoff that a(n) < 262*n^2 infinitely often, improving on an earlier result of Anderson & Stark. - Charles R Greathouse IV, Jun 14 2011
a(830) > 2 * 10^14 (probably around 3.511e14) and a(1160327) = 351753358289465 according to the calculations of Borwein, Ferguson, & Mossinghoff. - Charles R Greathouse IV, Jun 14 2011
a(n) is the smallest m such that A002819(m) = n.
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REFERENCES
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R. J. Anderson and H. M. Stark, Oscillation theorems, Analytic Number Theory (1980); Lecture Notes in Mathematics 899 (1981), pp. 79-106.
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LINKS
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EXAMPLE
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The sum of Liouville's function from 1 through 906150258 is 2, that is the smallest value, so a(2)=906150258.
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PROG
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(PARI) print1(r=1); t=0; for(n=906150257, 906400000, t+=(-1)^bigomega(n); if(t>r, r=t; print1(", "n))) \\ Charles R Greathouse IV, Jun 14 2011
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A002053
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a(n) = least value of m for which Liouville's function A002819(m) = -n.
(Formerly M0871 N0333)
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+10
7
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2, 3, 8, 13, 20, 31, 32, 53, 76, 79, 80, 117, 176, 181, 182, 193, 200, 283, 284, 285, 286, 293, 440, 443, 468, 661, 678, 683, 684, 1075, 1076, 1087, 1088, 1091, 1092, 1093, 1106, 1109, 1128, 1129, 1130, 1131, 1132, 1637, 1638, 1753, 1756, 1759, 1760, 2699
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OFFSET
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0,1
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COMMENTS
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Also when n first appears in A072203(m).
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REFERENCES
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]
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MATHEMATICA
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f[n_] := f[n] = f[n - 1] -(-1)^Length[Flatten[Table[ #[[1]], {#[[2]]}] & /@ FactorInteger[n]]]; f[1] = 0; Do[k = 1; While[f[k] != n, k++ ]; Print[k], {n, 1, 50}]
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PROG
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(Python)
from functools import reduce
from operator import ixor
from itertools import count
from sympy import factorint
def A002053(n): return next(filter(lambda m:-n==sum(-1 if reduce(ixor, factorint(i).values(), 0)&1 else 1 for i in range(1, m+1)), count(1))) # Chai Wah Wu, Jan 01 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A072203
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(Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).
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+10
4
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0, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 7, 8, 7, 6, 5, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 9, 8, 9, 10, 11, 10, 9, 10, 9, 8, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 2, 3, 4, 3, 4, 5, 6
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OFFSET
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1,3
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COMMENTS
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A number m is oddly or evenly factored depending on whether m has an odd or even number of prime factors, e.g., 12 = 2*2*3 has 3 factors so is oddly factored.
Polya conjectured that a(n) >= 0 for all n, but this was disproved by Haselgrove. Lehman gave the first explicit counterexample, a(906180359) = -1; the first counterexample is at 906150257 (Tanaka).
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REFERENCES
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G. Polya, Mathematics and Plausible Reasoning, S.8.16.
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LINKS
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FORMULA
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MATHEMATICA
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f[n_Integer] := Length[Flatten[Table[ #[[1]], {#[[2]]}] & /@ FactorInteger[n]]]; g[n_] := g[n] = g[n - 1] + If[ EvenQ[ f[n]], -1, 1]; g[1] = 0; Table[g[n], {n, 1, 103}]
Join[{0}, Accumulate[Rest[Table[If[OddQ[PrimeOmega[n]], 1, -1], {n, 110}]]]] (* Harvey P. Dale, Mar 10 2013 *)
Table[1 - Sum[(-1)^PrimeOmega[i], {i, 1, n}], {n, 1, 100}] (* Indranil Ghosh, Mar 17 2017 *)
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PROG
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(Haskell)
a072203 n = a072203_list !! (n-1)
a072203_list = scanl1 (\x y -> x + 2*y - 1) a066829_list
(PARI) a(n) = 1 - sum(i=1, n, (-1)^bigomega(i));
(Python)
from functools import reduce
from operator import ixor
from sympy import factorint
def A072203(n): return 1+sum(1 if reduce(ixor, factorint(i).values(), 0)&1 else -1 for i in range(1, n+1)) # Chai Wah Wu, Dec 20 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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Bill Dubuque (wgd(AT)zurich.ai.mit.edu), Jul 03 2002
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EXTENSIONS
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STATUS
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approved
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A175201
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a(n) is the smallest k such that the n consecutive values lambda(k), lambda(k+1), ..., lambda(k+n-1) = 1, where lambda(m) is the Liouville function A008836(m).
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+10
4
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1, 9, 14, 33, 54, 140, 140, 213, 213, 1934, 1934, 1934, 35811, 38405, 38405, 200938, 200938, 389409, 1792209, 5606457, 8405437, 8405437, 8405437, 8405437, 68780189, 68780189, 68780189, 68780189, 880346227, 880346227, 880346227, 880346227, 880346227
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OFFSET
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1,2
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COMMENTS
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Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo, where L(n) is the summatory Liouville function A002819(n). George Polya conjectured 1919 that L(n) <= 0 for all n >= 2. The conjecture was generally deemed true for nearly 40 years, until 1958, when C. B. Haselgrove proved that L(n) > 0 for infinitely many n. In 1962, R. S. Lehman found that L(906180359) = 1 and in 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257.
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REFERENCES
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
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LINKS
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FORMULA
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lambda(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.
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EXAMPLE
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a(1) = 1 and L(1) = 1;
a(2) = 9 and L(9) = L(10)= 1;
a(3) = 14 and L(14) = L(15) = L(16) = 1;
a(4) = 33 and L(33) = L(34) = L(35) = L(36) = 1.
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MAPLE
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with(numtheory):for k from 0 to 30 do : indic:=0:for n from 1 to 1000000000 while (indic=0)do :s:=0:for i from 0 to k do :if (-1)^bigomega(n+i)= 1 then s:=s+1: else fi:od:if s=k+1 and indic=0 then print(n):indic:=1:else fi:od:od:
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MATHEMATICA
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Table[k=1; While[Sum[LiouvilleLambda[k+i], {i, 0, n-1}]!=n, k++]; k, {n, 1, 30}]
With[{c=LiouvilleLambda[Range[841*10^4]]}, Table[SequencePosition[c, PadRight[ {}, n, 1], 1][[All, 1]], {n, 24}]//Flatten] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, Jul 27 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A189229
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Counterexamples to Polya's conjecture that A002819(n) <= 0 if n > 1.
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+10
3
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906150257, 906150258, 906150259, 906150260, 906150261, 906150262, 906150263, 906150264, 906150265, 906150266, 906150267, 906150268, 906150269, 906150270, 906150271, 906150272, 906150273, 906150274, 906150275, 906150276, 906150277, 906150278, 906150279, 906150280
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OFFSET
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1,1
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COMMENTS
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The point is that for all x < 906150257 there are more n <= x with Omega(n) odd than with Omega(n) even. At x = 906150257 the evens go ahead for the first time. - N. J. A. Sloane, Feb 10 2022
906150294 is the smallest number > 906150257 that is not in the sequence (see A028488).
See Brent and van de Lune (2011) for a history of Polya's conjecture and a proof that it is true "on average" in a certain precise sense.
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REFERENCES
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Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 22.
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LINKS
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FORMULA
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EXAMPLE
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906150257 is the smallest number k > 1 with A002819(k) > 0 (see Tanaka 1980).
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PROG
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(PARI) s=1; c=0; for(n=2, 906188859, s=s+(-1)^bigomega(n); if(s>0, c++; write("b189229.txt", c " " n))) /* Donovan Johnson, Apr 25 2013 */
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CROSSREFS
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Cf. A002819 (Liouville's summatory function L(n)), A008836 (Liouville's function lambda(n)), A028488 (n such that L(n) = 0), A051470 (least m for which L(m) = n).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A175202
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a(n) is the smallest k such that the n consecutive values L(k), L(k+1), ..., L(k+n-1) = -1, where L(m) is the Liouville function A008836(m).
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+10
2
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2, 2, 11, 17, 27, 27, 170, 279, 428, 5879, 5879, 13871, 13871, 13871, 41233, 171707, 1004646, 1004646, 1633357, 5460156, 11902755, 21627159, 21627159, 38821328, 41983357, 179376463, 179376463, 179376463, 179376463, 179376463, 179376463, 179376463
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OFFSET
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1,1
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COMMENTS
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L(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.
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REFERENCES
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H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
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LINKS
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EXAMPLE
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a(1) = 2 and L(2) = -1;
a(2) = 2 and L(2) = L(3)= -1;
a(3) = 11 and L(11) = L(12) = L(13) = -1;
a(4) = 17 and L(17) = L(18) = L(19) = L(20) = -1.
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MAPLE
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with(numtheory):for k from 0 to 30 do : indic:=0:for n from 1 to 1000000000 while (indic=0)do :s:=0:for i from 0 to k do :if (-1)^bigomega(n+i)= -1 then s:=s+1: else fi:od:if s=k+1 and indic=0 then print(n):indic:=1:else fi:od:od:
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MATHEMATICA
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Table[k=1; While[Sum[LiouvilleLambda[k+i], {i, 0, n-1}]!=-n, k++]; k, {n, 1, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A172357
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n such that the Liouville function lambda(n) take successively, from n, the values 1,-1,1,-1,1,-1
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+10
1
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58, 185, 194, 274, 287, 342, 344, 382, 493, 566, 667, 856, 858, 926, 1012, 1014, 1157, 1165, 1230, 1232, 1234, 1267, 1318, 1385, 1393, 1418, 1482, 1484, 1679, 1681, 1795, 1841, 1915, 1917, 2060, 2062, 2064, 2232, 2340, 2342, 2567, 2569, 2627, 2805, 3013
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OFFSET
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1,1
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LINKS
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MAPLE
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with(numtheory): for n from 1 to 4300 do; if (-1)^bigomega(n)=1 and (-1)^bigomega(n+1) = -1 and (-1)^bigomega(n+2) = 1 and (-1)^bigomega(n+3) = -1 and (-1)^bigomega(n+4) = 1 and (-1)^bigomega(n+5) = -1 then print(n); else fi ; od;
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MATHEMATICA
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Transpose[Transpose[#][[1]]&/@Select[Partition[Table[{n, LiouvilleLambda[ n]}, {n, 3100}], 6, 1], Transpose[#][[2]]=={1, -1, 1, -1, 1, -1}&]][[1]] Harvey P. Dale, May 19 2012
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PROG
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(PARI) lambda(n)=(-1)^bigomega(n);
for(n=1, 1e4, if(lambda(n)==1&lambda(n+1)==-1&lambda(n+2)==1&&lambda(n+3)==-1&lambda(n+4)==1&&lambda(n+5)==-1, print1(n", "))) /* Charles R Greathouse IV, Jun 13 2011 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A249482
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Numbers n such that the summatory Liouville function L(n) (A002819) is zero and L(n-1)*L(n+1) = -1.
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+10
1
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2, 906150256, 906150308, 906150310, 906151576, 906154582, 906154586, 906154604, 906154606, 906154608, 906154758, 906154762, 906154764, 906154768, 906154770, 906154788, 906154794, 906154824, 906154826, 906154828, 906154830, 906154836, 906154838, 906154856
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OFFSET
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1,1
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COMMENTS
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For k>=1,
in the interval [a(2k-1), a(2k)], L(n)<=0,
in the interval [a(2k), a(2k+1)], L(n)>=0.
In particular, for k=1, in the interval [2, 906150256], L(n)<=0.
G. Polya (1919) conjectured that L(n)<=0, for n>=2. But this was disproved in 1958 by B. Haselgrove, and in 1980 M. Tanaka found the smallest counterexample, a(2)+1 = 906150257.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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