A359791 Dirichlet inverse of function f(n) = 1 + A349905(n), where A349905(n) is the arithmetic derivative of prime shifted n.

1, -2, -2, -3, -2, -1, -2, -8, -7, -3, -2, 0, -2, -7, -5, -16, -2, 0, -2, -4, -9, -9, -2, 23, -11, -13, -40, -12, -2, 12, -2, -16, -11, -15, -11, 42, -2, -19, -15, 21, -2, 12, -2, -16, -24, -25, -2, 128, -19, -12, -17, -24, -2, 67, -13, 17, -21, -27, -2, 105, -2, -33, -48, 48, -17, 12, -2, -28, -27, 0, -2, 224

Offset: 1

Author

Antti Karttunen, Jan 13 2023

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003415, A003961, A349905, A359790.

Cf. A359764 (parity of terms), A359765 (positions of odd terms), A359766 (of even terms).

Cf. also A359169.

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));
memoA359791 = Map();
A359791(n) = if(1==n,1,my(v); if(mapisdefined(memoA359791,n,&v), v, v = -sumdiv(n,d,if(dA349905(n/d))*A359791(d),0)); mapput(memoA359791,n,v); (v)));

Formula

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

a(n) = A359790(A003961(n)).