A322358 Number of distinct twin prime pairs p, p+2 such that both of them divide n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset

1,105

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(n) = A001221(A322356(n)) = A001222(A322356(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A209328 = 0.107983... . - Amiram Eldar, Jan 01 2024

Example

For n = 45 = 3^2 * 5, there exists one twin prime pair (3,5) whose both members divide 45, thus a(45) = 1.
For n = 105 = 3 * 5 * 7, there exists two twin prime pairs, (3,5) and (5,7) whose both members divide 105, thus a(105) = 2.

Mathematica

f[p_, n_] := If[PrimeQ[p + 2] && Divisible[n, p*(p + 2)], 1, 0]; a[n_] := Plus @@ (f[#, n] & /@ FactorInteger[n][[;; , 1]]); Array[a, 105] (* Amiram Eldar, Dec 16 2018 *)

Prog

(PARI) A322358(n) = { my(ps=factor(n)[, 1]~); sum(i=1, #ps, isprime(ps[i]+2)*!(n%(ps[i]+2))); };

Crossrefs

Cf. A001359, A006512, A209328, A322356.

Sequence in context: A235145 A266342 A285936 * A322437 A330174 A336835

Adjacent sequences: A322355 A322356 A322357 * A322359 A322360 A322361

Keyword

nonn

Author

Antti Karttunen, Dec 16 2018

Status

approved