A355829 Dirichlet inverse of A009194, the greatest common divisor of sigma(n) and n, where sigma is the sum of divisors function.

1, -1, -1, 0, -1, -4, -1, 0, 0, 0, -1, 7, -1, 0, -1, 0, -1, 8, -1, 1, 1, 0, -1, -10, 0, 0, 0, -25, -1, 10, -1, 0, -1, 0, 1, 15, -1, 0, 1, -8, -1, 6, -1, -1, 2, 0, -1, 16, 0, 2, -1, 1, -1, -6, 1, 46, 1, 0, -1, -9, -1, 0, 0, 0, 1, 10, -1, 1, -1, 2, -1, -29, -1, 0, 4, -1, 1, 6, -1, 16, 0, 0, -1, 29, 1, 0, -1, 2, -1, -8

Offset: 1

Author

Antti Karttunen, Jul 20 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A009194.

Cf. also A355828.

Programs

Mathematica

s[n_] := GCD[n, DivisorSigma[1, n]]; a[1] = 1; a[n_] := - DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 20 2022 *)

PARI

A009194(n) = gcd(n, sigma(n));
memoA355829 = Map();
A355829(n) = if(1==n,1,my(v); if(mapisdefined(memoA355829,n,&v), v, v = -sumdiv(n,d,if(dA009194(n/d)*A355829(d),0)); mapput(memoA355829,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA009194(n/d) * a(d).