A359589 Dirichlet inverse of function f(n) = (-1 + gcd(A003415(n), A276086(n))), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

1, 0, 0, 0, 0, -4, 0, -2, -5, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, -2, -9, 0, 0, 0, -9, -14, -2, 0, 0, 0, 0, 0, -13, 0, -5, 12, 0, -20, -1, 0, 0, 0, 0, -2, -2, -24, 0, 10, -13, -14, -9, -6, 0, 40, -1, 0, -1, 0, 0, 0, 0, -2, -2, 2, -17, 0, 0, -2, -1, 0, 0, 20, 0, -2, -4, -4, -17, 0, 0, 0, 20, 0, 0, 16, -1, -14, -1, -34

Offset: 1

Author

Antti Karttunen, Jan 09 2023

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030

Crossrefs

Cf. A003415, A276086, A327858, A359595 (parity of terms), A359596 (positions of odd terms).

Agrees paritywise with A358777.

Cf. also A346242, A354347, A354348, A359603.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n), A276086(n));
memoA359589 = Map();
A359589(n) = if(1==n,1,my(v); if(mapisdefined(memoA359589,n,&v), v, v = -sumdiv(n,d,if(dA327858(n/d)-1)*A359589(d),0)); mapput(memoA359589,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA327858(n/d)-1) * a(d).

a(n) == A358777(n) mod 2.