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

1,3

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

Formula

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

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

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 *)

Prog

(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));

Crossrefs

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

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

Sequence in context: A335965 A225176 A349271 * A118665 A333238 A336526

Adjacent sequences: A349384 A349385 A349386 * A349388 A349389 A349390

Keyword

nonn,mult

Author

Antti Karttunen, Nov 17 2021

Status

approved