Displaying 1-10 of 12 results found.
a(n) = least value of m for which Liouville's function A002819(m) = -n.
(Formerly M0871 N0333)
+0
7
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
COMMENTS
Also when n first appears in A072203(m).
REFERENCES
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).
LINKS
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]
MATHEMATICA
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}]
PROG
(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
Liouville's function L(n) = partial sums of A008836.
(Formerly M0042 N0012)
+0
31
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
COMMENTS
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
REFERENCES
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).
LINKS
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.
MATHEMATICA
Accumulate[Join[{0}, LiouvilleLambda[Range[90]]]] (* Harvey P. Dale, Nov 08 2011 *)
PROG
(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
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jul 09 2001
Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).
+0
191
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
COMMENTS
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
REFERENCES
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.
LINKS
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]
FORMULA
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)
EXAMPLE
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.
MAPLE
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:
MATHEMATICA
Table[If[OddQ[PrimeOmega[n]], -1, 1], {n, 110}] (* Harvey P. Dale, Sep 10 2014 *)
PROG
(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
CROSSREFS
Cf. A000005, A001222, A002053, A007421, A002819 (partial sums), A008683, A010052, A026424, A028260, A028488, A056912, A056913, A065043, A066829, A106400, A156552, A349905, A063647, A101455, A048691.
a(n) is least value of m for which the sum of Liouville's function from 1 to m is n.
+0
12
1, 906150258, 906150259, 906150260, 906150263, 906150264, 906150331, 906150334, 906150337, 906150338, 906150339, 906150358, 906150359, 906150362, 906150363, 906150368, 906150387, 906150388, 906150389, 906150406, 906150407
COMMENTS
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.
REFERENCES
R. J. Anderson and H. M. Stark, Oscillation theorems, Analytic Number Theory (1980); Lecture Notes in Mathematics 899 (1981), pp. 79-106.
EXAMPLE
The sum of Liouville's function from 1 through 906150258 is 2, that is the smallest value, so a(2)=906150258.
PROG
(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
Discriminants of real quadratic fields with class number 2 and related continued fraction period length of 5.
+0
0
74, 218, 493, 565, 1037, 1565, 1781, 2138, 2165, 2173, 3869, 5165, 5213, 5837, 6485, 8021, 10397, 14213
REFERENCES
R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A3.
(Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).
+0
4
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
COMMENTS
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).
REFERENCES
G. Polya, Mathematics and Plausible Reasoning, S.8.16.
MATHEMATICA
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 *)
PROG
(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
AUTHOR
Bill Dubuque (wgd(AT)zurich.ai.mit.edu), Jul 03 2002
n such that the Liouville function lambda(n) take successively, from n, the values 1,-1,1,-1,1,-1
+0
1
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
MAPLE
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;
MATHEMATICA
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
PROG
(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 */
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).
+0
4
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
COMMENTS
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.
REFERENCES
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.
FORMULA
lambda(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.
EXAMPLE
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.
MAPLE
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:
MATHEMATICA
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 *)
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).
+0
2
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
COMMENTS
L(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.
REFERENCES
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.
EXAMPLE
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.
MAPLE
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:
MATHEMATICA
Table[k=1; While[Sum[LiouvilleLambda[k+i], {i, 0, n-1}]!=-n, k++]; k, {n, 1, 30}]
Counterexamples to Polya's conjecture that A002819(n) <= 0 if n > 1.
+0
3
906150257, 906150258, 906150259, 906150260, 906150261, 906150262, 906150263, 906150264, 906150265, 906150266, 906150267, 906150268, 906150269, 906150270, 906150271, 906150272, 906150273, 906150274, 906150275, 906150276, 906150277, 906150278, 906150279, 906150280
COMMENTS
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.
REFERENCES
Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 22.
EXAMPLE
906150257 is the smallest number k > 1 with A002819(k) > 0 (see Tanaka 1980).
PROG
(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 */
CROSSREFS
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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