A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

2, 0, 0, 1, 0, 4, 0, 9, 4, 4, 0, 32, 0, 8, 8, 49, 0, 56, 0, 36, 16, 4, 0, 153, 4, 8, 56, 66, 0, 96, 0, 207, 8, 4, 16, 295, 0, 8, 16, 187, 0, 168, 0, 48, 120, 12, 0, 553, 16, 80, 8, 78, 0, 444, 8, 323, 16, 4, 0, 480, 0, 12, 216, 745, 16, 144, 0, 60, 24, 200, 0, 1016, 0, 8, 152, 90, 16, 216, 0, 723, 472, 4, 0, 786, 8, 8, 8, 289

Offset: 1

Author

Antti Karttunen, Aug 19 2021

Keywords

• sign

Comments

Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.

The first negative term is a(720) = -6306.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for primes, gaps between
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A001223, A001248, A003961, A063524, A336853, A347098.

Cf. also A346250, A347097.

Programs

PARI

up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];
A347099(n) = if(1==n,2,A336853(n)+A347098(n));

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A336853(d) * A347098(n/d).

For all n >= 1, a(A001248(n)) = A001223(n)^2.