A347097 a(1) = 2; and for n > 1, a(n) = A341512(n) + A347096(n).

2, 0, 0, 1, 0, 4, 0, 21, 4, 4, 0, 110, 0, 8, 8, 259, 0, 224, 0, 154, 16, 4, 0, 1548, 4, 8, 176, 316, 0, 592, 0, 2445, 8, 4, 16, 4312, 0, 8, 16, 2450, 0, 1216, 0, 382, 640, 12, 0, 15532, 16, 408, 8, 616, 0, 6708, 8, 5064, 16, 4, 0, 12272, 0, 12, 1312, 19543, 16, 1504, 0, 754, 24, 1568, 0, 50561, 0, 8, 832, 1060, 16

Offset: 1

Author

Antti Karttunen, Aug 19 2021

Keywords

• sign

Comments

Sum of {the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.

The first negative term is a(5760) = -1223227750.

Links

• Index entries for primes, gaps between
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A001223, A001248, A003961, A063524, A341512, A347096.

Cf. also A346236, A346240, A347099.

Programs

PARI

up_to = 16384;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA341512(n));
v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
A347096(n) = v347096[n];
A347097(n) = if(1==n,2,A341512(n) + A347096(n));

Formula

a(1) = 2, and for n>1, a(n) = -Sum_{d|n, 1A341512(d) * A347096(n/d).

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