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.

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

Offset: 1

Author

Ilya Gutkovskiy, Jul 19 2021

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..3223

Crossrefs

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346456.

Programs

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 *)

PARI

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

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.