A359169 Dirichlet inverse of the pointwise sum of A349905 (arithmetic derivative of prime shifted n) and A063524 (1, 0, 0, 0, ...).

1, -1, -1, -5, -1, -6, -1, -16, -9, -8, -1, -14, -1, -12, -10, -35, -1, -22, -1, -22, -14, -14, -1, 10, -13, -18, -56, -38, -1, -17, -1, -31, -16, -20, -16, 42, -1, -24, -20, -2, -1, -33, -1, -46, -54, -30, -1, 243, -21, -46, -22, -62, -1, -10, -18, -26, -26, -32, -1, 140, -1, -38, -86, 160, -22, -41, -1, -70, -32, -53

Offset: 1

Author

Antti Karttunen, Dec 23 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003415, A003961, A346241, A349905.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349905(n) = A003415(A003961(n));
memoA359169 = Map();
A359169(n) = if(1==n,1,my(v); if(mapisdefined(memoA359169,n,&v), v, v = -sumdiv(n,d,if(dA349905(n/d)*A359169(d),0)); mapput(memoA359169,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA349905(n/d) * a(d).

a(n) = A346241(A003961(n)).