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Search: a051470 -id:a051470
Displaying 1-10 of 11 results found. page 1 2
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A028488 Numbers k such that the summatory Liouville function L(k) (A002819) is zero. +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
All terms are even since k and A002819(k) have the same parity. - Jianing Song, Aug 06 2021
LINKS
Donovan Johnson and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..317312 (a(1)-a(252) from Donovan Johnson)
P. Borwein, R. Ferguson, and M. Mossinghoff, Sign changes in sums of the Liouville function, Mathematics of Computation 77 (2008), pp. 1681-1694.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3, 187-189, 1980.
Hiroaki Yamanouchi, Values of L(n) from 2*10^14 to 3.75*10^14 (interval = 5*10^9)
Eric Weisstein's World of Mathematics, Liouville Function
Eric Weisstein's World of Mathematics, Polya Conjecture
MAPLE
B:= [seq((-1)^numtheory:-bigomega(i), i=1..10^5)]:
L:= ListTools:-PartialSums(B):
select(t -> L[t]=0, [$1..10^5]); # Robert Israel, Aug 27 2015
MATHEMATICA
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 *)
CROSSREFS
Cf. A008836 (Liouville's function), A002819, A051470.
KEYWORD
nonn,nice
AUTHOR
Eric W. Weisstein
EXTENSIONS
More terms from Hans Havermann, Jun 24 2002
STATUS
approved
A072203 (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n). +10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
C. B. Haselgrove, A disproof of a conjecture of Polya, Mathematika 5 (1958), pp. 141-145.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Kyle Sturgill-Simon, An interesting opportunity: the Gilbreath conjecture, Honors Thesis, Mathematics Dept., Carroll College, 2012.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3:1, 187-189, 1980.
FORMULA
a(n) = 1 - A002819(n). - T. D. Noe, Feb 06 2007
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
-- Reinhard Zumkeller, Nov 19
(PARI) a(n) = 1 - sum(i=1, n, (-1)^bigomega(i));
for(n=1, 100, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 17 2017
(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
CROSSREFS
Cf. A028488, A002819, A051470, A066829.
KEYWORD
sign,nice,easy,look
AUTHOR
Bill Dubuque (wgd(AT)zurich.ai.mit.edu), Jul 03 2002
EXTENSIONS
Edited and extended by Robert G. Wilson v, Jul 13 2002
Comment corrected by Charles R Greathouse IV, Mar 08 2010
STATUS
approved
A175201 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). +10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
LINKS
Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..44 (terms < 10^13, first 37 terms from Donovan Johnson)
Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
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 *)
CROSSREFS
Cf. A172354, A051470, A028488, A002819, A175202, A066793.
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 04 2010
STATUS
approved
A090410 Values of L(10^n), where L(n) is the summatory function of the Liouville function A008836(n). +10
3
1, 0, -2, -14, -94, -288, -530, -842, -3884, -25216, -116026, -342224, -522626, -966578, -7424752, -29445104, -97617938, -271676470, -618117940, -810056106 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Table of n, a(n) for n=0..19.
Eric Weisstein's World of Mathematics, Liouville Function
FORMULA
a(n) = A002819(10^n). - Ray Chandler, May 30 2012
PROG
(PARI) a(n) = sum(i=1, 10^n, (-1)^bigomega(i)); \\ Michel Marcus, Sep 29 2015
CROSSREFS
Cf. A002819, A008836.
KEYWORD
sign,more
AUTHOR
Eric W. Weisstein, Nov 30 2003
EXTENSIONS
a(9)-a(13) from Donovan Johnson, Mar 08 2011
a(14)-a(17) from Hiroaki Yamanouchi, Jul 13 2014
a(18) from Henri Lifchitz, Dec 01 2014
a(19) from Hiroaki Yamanouchi, Sep 28 2015
STATUS
approved
A189229 Counterexamples to Polya's conjecture that A002819(n) <= 0 if n > 1. +10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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 A002819, A008836, A028488, A051470 for additional comments, references, and links.
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.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..10000
R. P. Brent and J. van de Lune, A note on Polya's observation concerning Liouville's function, arXiv:1112.4911 [math.NT] 2011.
Jarosław Grytczuk, From the 1-2-3 Conjecture to the Riemann Hypothesis, arXiv:2003.02887 [math.CO], 2020. See p. 9.
Ben Sparks, 906,150,257 and the Pólya conjecture (MegaFavNumbers), SparksMath video (2020).
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3 (1980), 187-189.
Wikipedia, Pólya conjecture.
FORMULA
{ k : (k-1)*A002819(k) > 0. }
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).
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jun 13 2011
STATUS
approved
A346457 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. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..3223
FORMULA
a(n) = min {k : |Sum_{j=1..k} mu(rad(j))| = n}, where mu is the Moebius function and rad is the squarefree kernel.
MAPLE
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:
convert(V, list); # Robert Israel, Oct 30 2023
MATHEMATICA
Table[k=1; While[Abs[Sum[(-1)^PrimeNu@j, {j, k}]]!=n, k++]; k, {n, 30}] (* Giorgos Kalogeropoulos, Jul 19 2021 *)
PROG
(PARI) a(n) = my(k=1); while (abs(sum(j=1, k, (-1)^omega(j))) != n, k++); k; \\ Michel Marcus, Jul 19 2021
CROSSREFS
Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346456.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 19 2021
STATUS
approved
A175202 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). +10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
LINKS
Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..43 (terms < 10^13, first 38 terms from Donovan Johnson)
Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function. Math. Comp. 77 (2008), 1681-1694.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
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}]
CROSSREFS
Cf. A002819, A008836, A028488, A051470, A066794, A172354, A175201.
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 04 2010
EXTENSIONS
a(15) and a(21) corrected by Donovan Johnson, Apr 01 2013
STATUS
approved
A346455 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. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..60.
FORMULA
a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = n}, where mu is the Moebius function and rad is the squarefree kernel.
MATHEMATICA
a[n_]:=(k=1; While[Sum[(-1)^PrimeNu@j, {j, k}]!=n, k++]; k); Array[a, 25] (* Giorgos Kalogeropoulos, Jul 19 2021 *)
PROG
(PARI) a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) !=n, k++); k; \\ Michel Marcus, Jul 19 2021
CROSSREFS
Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346456, A346457.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 19 2021
STATUS
approved
A346456 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. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..48.
FORMULA
a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = -n}, where mu is the Moebius function and rad is the squarefree kernel.
MATHEMATICA
a[n_]:=(k=1; While[Sum[(-1)^PrimeNu@j, {j, k}]!=-n, k++]; k); Array[a, 6] (* Giorgos Kalogeropoulos, Jul 19 2021 *)
PROG
(PARI) a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) != -n, k++); k; \\ Michel Marcus, Jul 19 2021
CROSSREFS
Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346457.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 19 2021
STATUS
approved
A172357 n such that the Liouville function lambda(n) take successively, from n, the values 1,-1,1,-1,1,-1 +10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
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 */
CROSSREFS
Cf. A051470, A028488, A002819.
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 01 2010
STATUS
approved
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