A355828 Dirichlet inverse of A342671, the greatest common divisor of sigma(n) and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, -3, -1, 8, -1, 3, -1, -24, 0, 3, -1, -8, -1, 3, 1, 72, -1, 0, -1, -28, 1, 3, -1, 12, 0, 3, -4, -8, -1, -3, -1, -222, 1, 3, 1, 0, -1, 3, 1, 138, -1, -3, -1, -10, 0, 3, -1, 0, 0, 0, 1, -8, -1, 12, 1, 24, -3, 3, -1, 28, -1, 3, 0, 684, -5, -3, -1, -16, 1, -3, -1, 12, -1, 3, 0, -8, 1, -3, -1, -538, 8, 3, -1, 8, 1, 3, -3, 30

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 computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A342671.

Cf. also A355829.

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; s[n_] := GCD[DivisorSigma[1, n], Times @@ f @@@ FactorInteger[n]]; a[1] = 1; a[n_] := - DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 20 2022 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
memoA355828 = Map();
A355828(n) = if(1==n,1,my(v); if(mapisdefined(memoA355828,n,&v), v, v = -sumdiv(n,d,if(dA342671(n/d)*A355828(d),0)); mapput(memoA355828,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA342671(n/d) * a(d).