A354823 Dirichlet inverse of A351083, where A351083(n) = gcd(n, A327860(n)), and A327860 is the arithmetic derivative of the primorial base exp-function.

1, -1, -1, -1, -1, 1, -7, -5, 0, 1, -1, 1, -1, 13, -3, -1, -1, -2, -1, -7, 13, 1, -1, 9, -24, 1, 0, 7, -1, 7, -1, 33, 1, -15, 9, -6, -1, 1, -11, 27, -1, -25, -1, -1, 4, 1, -1, 7, 48, 24, 1, -1, -1, 2, -3, 59, 1, 1, -1, 19, -1, 1, -12, 23, 1, -1, -1, 33, 1, -23, -1, -2, -1, 1, 52, 1, 7, 23, -1, -67, 0, 1, -1, -25

Offset: 1

Author

Antti Karttunen, Jun 09 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. A327860, A351083.

Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).

Cf. also A346242, A354347, A354348, A354824.

Programs

PARI

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A351083(n) = gcd(n, A327860(n));
memoA354823 = Map();
A354823(n) = if(1==n,1,my(v); if(mapisdefined(memoA354823,n,&v), v, v = -sumdiv(n,d,if(dA351083(n/d)*A354823(d),0)); mapput(memoA354823,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA351083(n/d) * a(d).