A359790 Dirichlet inverse of function f(n) = 1 + n', where n' stands for the arithmetic derivative of n, A003415(n).

1, -2, -2, -1, -2, 2, -2, -1, -3, 0, -2, 3, -2, -2, -1, 0, -2, 6, -2, 3, -3, -6, -2, 7, -7, -8, -8, 3, -2, 12, -2, 3, -7, -12, -5, 9, -2, -14, -9, 11, -2, 18, -2, 3, 0, -18, -2, 11, -11, 6, -13, 3, -2, 26, -9, 15, -15, -24, -2, 17, -2, -26, -4, 9, -11, 30, -2, 3, -19, 16, -2, 9, -2, -32, 0, 3, -11, 36, -2, 23, -16, -36

Offset: 1

Author

Antti Karttunen, Jan 13 2023

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A359780, A359781 (parity of terms), A359782 (positions of even terms), A359783 (of odd terms).

Cf. also A346241, A347082, A347084, A359603, A359789, A359791 [= a(A003961(n))] (for similar constructions).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
memoA359790 = Map();
A359790(n) = if(1==n,1,my(v); if(mapisdefined(memoA359790,n,&v), v, v = -sumdiv(n,d,if(dA003415(n/d))*A359790(d),0)); mapput(memoA359790,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA003415(n/d)) * a(d).