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
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
KEYWORD
nonn
AUTHOR
Mats Granvik, Mar 03 2019
STATUS
approved