Search: a051470 -id:a051470
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A028488
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Numbers k such that the summatory Liouville function L(k) (A002819) is zero.
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+10
13
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2, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, 906150294, 906150308, 906150310, 906150314, 906151516, 906151576, 906152172, 906154582, 906154586, 906154590, 906154594, 906154604, 906154606, 906154608, 906154758, 906154760, 906154762
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OFFSET
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1,1
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COMMENTS
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a(253) > 2*10^14 according to the calculations of Borwein, Ferguson, & Mossinghoff. Most likely a(253) = 351100332278250. - Charles R Greathouse IV, Jun 14 2011
L(23156358837978983978) = 0 and L(k) < 0 for k from 2.3156354*10^19 to 23156358837978983977. - Hiroaki Yamanouchi, Oct 03 2015
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LINKS
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MAPLE
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B:= [seq((-1)^numtheory:-bigomega(i), i=1..10^5)]:
L:= ListTools:-PartialSums(B):
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MATHEMATICA
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Position[Table[Sum[LiouvilleLambda@ k, {k, 1, n}], {n, 1000}], n_ /; n == 0] // Flatten (* Michael De Vlieger, Aug 27 2015 *)
Position[Accumulate[LiouvilleLambda[Range[1000]]], 0]//Flatten (* Harvey P. Dale, Aug 10 2022 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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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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A090410
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Values of L(10^n), where L(n) is the summatory function of the Liouville function A008836(n).
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+10
3
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1, 0, -2, -14, -94, -288, -530, -842, -3884, -25216, -116026, -342224, -522626, -966578, -7424752, -29445104, -97617938, -271676470, -618117940, -810056106
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OFFSET
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0,3
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COMMENTS
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L(n) for n <= 10^13 is always negative from 906488081 to 10^13. It reaches a record negative value of -3458310 at 8196557476890. It reaches a record positive value of 829 at 906316571 (A051470(829)). - Donovan Johnson, Mar 08 2011
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LINKS
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FORMULA
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PROG
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(PARI) a(n) = sum(i=1, 10^n, (-1)^bigomega(i)); \\ Michel Marcus, Sep 29 2015
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CROSSREFS
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KEYWORD
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sign,more
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AUTHOR
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EXTENSIONS
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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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A346457
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a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.
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+10
3
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1, 4, 5, 8, 9, 32, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = min {k : |Sum_{j=1..k} mu(rad(j))| = n}, where mu is the Moebius function and rad is the squarefree kernel.
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MAPLE
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N:= 10000: # for values <= N
omega:= n -> nops(numtheory:-factorset(n)):
R:= map(n -> (-1)^omega(n), [$1..10000]):
S:= map(abs, ListTools:-PartialSums(R)):
m:= max(S):
V:= Vector(m):
for i from 1 to N do if S[i] > 0 and V[S[i]] = 0 then V[S[i]]:= i fi od:
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MATHEMATICA
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Table[k=1; While[Abs[Sum[(-1)^PrimeNu@j, {j, k}]]!=n, k++]; k, {n, 30}] (* Giorgos Kalogeropoulos, Jul 19 2021 *)
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PROG
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(PARI) a(n) = my(k=1); while (abs(sum(j=1, k, (-1)^omega(j))) != n, k++); k; \\ Michel Marcus, Jul 19 2021
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CROSSREFS
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Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346456.
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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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A346455
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a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = n, where omega(j) is the number of distinct primes dividing j.
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+10
2
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1, 52, 55, 56, 57, 58, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = n}, where mu is the Moebius function and rad is the squarefree kernel.
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MATHEMATICA
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a[n_]:=(k=1; While[Sum[(-1)^PrimeNu@j, {j, k}]!=n, k++]; k); Array[a, 25] (* Giorgos Kalogeropoulos, Jul 19 2021 *)
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PROG
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(PARI) a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) !=n, k++); k; \\ Michel Marcus, Jul 19 2021
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CROSSREFS
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Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346456, A346457.
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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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A346456
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a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = -n, where omega(j) is the number of distinct primes dividing j.
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+10
2
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3, 4, 5, 8, 9, 32, 9283, 9284, 9285, 9292, 9293, 9294, 9295, 9296, 9343, 9434, 9437, 9440, 9479, 9686, 9689, 9690, 9697, 9698, 9699, 9700, 9711, 9716, 9717, 9718, 9719, 9720, 9721, 9740, 9741, 9852, 9855, 9856, 9857, 10284, 10285, 10286, 10305, 10314, 10325, 10326, 10331, 10338
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = -n}, where mu is the Moebius function and rad is the squarefree kernel.
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MATHEMATICA
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a[n_]:=(k=1; While[Sum[(-1)^PrimeNu@j, {j, k}]!=-n, k++]; k); Array[a, 6] (* Giorgos Kalogeropoulos, Jul 19 2021 *)
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PROG
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(PARI) a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) != -n, k++); k; \\ Michel Marcus, Jul 19 2021
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CROSSREFS
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Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346457.
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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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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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