A378606 Dirichlet convolution of A046692 and A003961, where A046692 is the Dirichlet inverse of sigma, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

1, 0, 1, 2, 1, 0, 3, 6, 8, 0, 1, 2, 3, 0, 1, 18, 1, 0, 3, 2, 3, 0, 5, 6, 12, 0, 40, 6, 1, 0, 5, 54, 1, 0, 3, 16, 3, 0, 3, 6, 1, 0, 3, 2, 8, 0, 5, 18, 40, 0, 1, 6, 5, 0, 1, 18, 3, 0, 1, 2, 5, 0, 24, 162, 3, 0, 3, 2, 5, 0, 1, 48, 5, 0, 12, 6, 3, 0, 3, 18, 200, 0, 5, 6, 1, 0, 1, 6, 7, 0, 9, 10, 5, 0, 3, 54, 3, 0, 8, 24

Offset: 1

Author

Antti Karttunen, Dec 11 2024

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to prime indices in the factorization of n.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A003961, A008683, A016825 (positions of 0's), A046692, A151800, A349387 (inverse Möbius transform), A378607 (Dirichlet inverse).

Programs

Mathematica

f[p_, e_] := Module[{q = NextPrime[p]}, If[e == 1, q - p - 1, q^e - (p + 1)*q^(e - 1) + p*q^(e - 2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 11 2024 *)

PARI

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
A378606(n) = sumdiv(n,d,A046692(d)*A003961(n/d));

Formula

a(n) = Sum_{d|n} A046692(d)*A003961(n/d).

a(n) = Sum_{d|n} A008683(d)*A349387(n/d).

Multiplicative with a(p^e) = q(p)^e - (p+1) * q(p)^(e-1) + p * q(p)^(e-2) if e >= 2, and q(p) - p - 1 if e = 1, where q(p) = A151800(p) is the prime next to p. - Amiram Eldar, Dec 11 2024