A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

1, -1, -1, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -9, -1, -2, -1, 1, -3, 1, -1, 1, -4, -3, 0, 1, -1, -1, -1, 21, 1, 1, -1, -6, -1, 1, 1, 3, -1, 7, -1, -1, 0, -3, -1, 23, 0, 4, -3, 7, -1, 2, 1, 3, 1, 1, -1, -1, -1, 1, 8, 15, -1, -1, -1, 1, 1, 3, -1, 14, -1, 1, -46, -7, -1, 7, -1, 5, 0, 1, -1, 3, 1, -3, 1, -131

Offset: 1

Author

Antti Karttunen, Jun 07 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030
• Index entries for sequences related to primorial base

Crossrefs

Cf. A003415, A276086, A345000.

Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).

Cf. also A346242, A354348, A354823, A354824.

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); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
memoA354347 = Map();
A354347(n) = if(1==n,1,my(v); if(mapisdefined(memoA354347,n,&v), v, v = -sumdiv(n,d,if(dA345000(n/d)*A354347(d),0)); mapput(memoA354347,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA345000(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A353627(n).