A349387 Dirichlet convolution of A003961 with A055615 (Dirichlet inverse of n), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, 1, 2, 3, 2, 2, 4, 9, 10, 2, 2, 6, 4, 4, 4, 27, 2, 10, 4, 6, 8, 2, 6, 18, 14, 4, 50, 12, 2, 4, 6, 81, 4, 2, 8, 30, 4, 4, 8, 18, 2, 8, 4, 6, 20, 6, 6, 54, 44, 14, 4, 12, 6, 50, 4, 36, 8, 2, 2, 12, 6, 6, 40, 243, 8, 4, 4, 6, 12, 8, 2, 90, 6, 4, 28, 12, 8, 8, 4, 54, 250, 2, 6, 24, 4, 4, 4, 18, 8, 20, 16, 18, 12, 6

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• nonn
• mult

Comments

Multiplicative because A003961 and A055615 are.

Convolving this with A000010 gives A003972, and convolving this with A000203 gives A003973.

Multiplicative with a(p^e) = nextprime(p)^e - p * nextprime(p)^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

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. A000040, A001223, A003961, A055615, A151800, A349388 (Dirichlet inverse), A349389 (sum with it), A378606 (Möbius transform).

Cf. also A000010, A000203, A003972, A003973, A347236, A349381.

Programs

Mathematica

f[p_,e_] := (q = NextPrime[p])^e - p * q^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A055615(n) = (n*moebius(n));
A349387(n) = sumdiv(n,d,A003961(n/d)*A055615(d));

Formula

a(n) = Sum_{d|n} A003961(n/d) * A055615(d).

For all n >= 1, a(A000040(n)) = A001223(n).