A306652 a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*[k divides m + 4].

1, 2, 2, 4, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 4, 10, 2, 4, 2, 8, 4, 4, 2, 12, 2, 4, 2, 8, 2, 8, 2, 14, 4, 4, 4, 8, 2, 4, 4, 12, 2, 8, 2, 8, 4, 4, 2, 20, 2, 4, 4, 8, 2, 4, 4, 12, 4, 4, 2, 16, 2, 4, 4, 14, 4, 8, 2, 8, 4, 8, 2, 12, 2, 4, 4, 8, 4, 8, 2, 20, 2, 4, 2, 16, 4

Offset: 1

Author

Mats Granvik, Mar 03 2019

Keywords

• nonn

Comments

According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes. In an analogy of Dirichlet convolutions A147848 corresponds to the twin primes while this sequence corresponds to the cousin primes, and it appears that this sequence differs from A147848 at multiples of 16. See A306653 for comparison.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

Crossrefs

Cf. A147848, A306652, A298825, A008683.

Programs

Mathematica

a[n_] := Sum[Sum[If[Mod[n, k] == 0, If[Mod[m, n/k] == 0, 1, 0], 0]*If[Mod[m + 4, k/1] == 0, 1, 0], {k, 1, n}], {m, 1, n}]; a /@Range[85] (* Dirichlet Convolution. *)
a[n_] := Sum[If[Mod[n, k] == 0, Sum[If[Mod[4, j] == 0, If[GCD[k, n/k] == j, j, 0], 0], {j, 1, n}], 0], {k, 1, n}]; a /@Range[85] (* GCD sum. *)
a[n_] := Sum[If[Mod[4, j] == 0, j*Count[Divisors[n], d_ /; GCD[d, n/d] == j], 0], {j, 1, n}]; a /@Range[85] (* After Jean-François Alcover in A034444. *)

PARI

A306652(n) = sum(m=1, n, sumdiv(n, k, !(m%(n/k)) && !((m+4)%k))); \\ Antti Karttunen, Mar 13 2019

Formula

a(n) = Sum_{m=1..n} Sum_{k=1..n} [k divides n]*[n/k divides m]*[k divides m + 4].

a(n) = Sum_{k=1..n}[k divides n]*Sum_{j=1..n}[j divides 4]*[GCD[k, n/k] = j]*j.