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

1,2

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

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.

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. *)

Prog

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

Crossrefs

Cf. A147848, A306652, A298825, A008683.

Sequence in context: A092520 A372714 A147848 * A239614 A358216 A193432

Adjacent sequences: A306649 A306650 A306651 * A306653 A306654 A306655

Keyword

nonn

Author

Mats Granvik, Mar 03 2019

Status

approved