A051402 -id:A051402 - cp's OEIS frontend

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.

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

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346457.

a[n_]:=(k=1;While[Sum[(-1)^PrimeNu@j,{j,k}]!=-n,k++];k);Array[a,6] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) != -n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = -n}, where mu is the Moebius function and rad is the squarefree kernel.

A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

Original entry on oeis.org

37, 1, 18, 7, 28, 8, 44, 4, 1, 9, 1, 3, 1, 2, 48, 17, 1, 3, 1, 2, 16, 75, 2, 1, 1, 20, 2, 1, 2, 4, 1, 1, 2, 27, 8, 2, 1, 1, 2, 1, 5, 1, 5, 1, 2, 1, 1, 1, 2, 1, 109, 4, 66, 1, 27, 1, 1, 144, 4, 8, 2, 1, 2, 13, 1, 2, 9, 1, 1, 24, 1, 3, 16, 8, 6, 1, 2, 3, 4, 2, 1, 2, 5, 1, 2, 4, 3, 2, 1, 3, 1, 82, 3, 5

Offset: 1

Labos Elemer, Oct 11 2000

nonn

Mertens's function (A002321) is oscillating. The width of its waves is given here.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002321, A013928, A028442, A013940, A051400, A051401, A051402.

Differences[Position[Accumulate[Array[MoebiusMu,1500]],0]//Flatten] (* Harvey P. Dale, Nov 10 2016 *)

lista(kmax) = {my(s = 0, k1 = 2); for(k2 = 3, kmax, s += moebius(k2); if(s == 0, print1(k2 - k1, ", "); k1 = k2));} \\ Amiram Eldar, Jun 09 2024

a(n) = A028442(n+1) - A028442(n).

Offset corrected by Amiram Eldar, Jun 09 2024

A084235 Smallest k such that |M(k)| = 2^n, where M(x) is Mertens's function A002321.

Original entry on oeis.org

1, 5, 31, 199, 1637, 8507, 24110, 95961, 355541, 1786062, 6473791, 30649362, 109851909, 456774199, 2589994747, 7613644886, 37725066313, 108500046711

Offset: 1

Robert G. Wilson v, May 13 2003

Karl Sabbagh, The Riemann Hypothesis, The Greatest Unsolved Problem in Mathematics, Farrar, Straus and Giroux, New York, 2002, page 191.

Cf. A002321, A051402, A084234.

i = s = 0; Do[ While[ Abs[s] < 2^n, s = s + MoebiusMu[i]; i++ ]; Print[i - 1], {n, 0, 25}]

a(n) = A051402(2^n). - Amiram Eldar, May 06 2024

Definition corrected by L. Edson Jeffery, Mar 18 2013

a(15)-a(18) by Ashley Jordan and Luke March, Jul 22 2014

cp's OEIS Frontend

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.

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A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

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A084235 Smallest k such that |M(k)| = 2^n, where M(x) is Mertens's function A002321.

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