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

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Offset: 1

Author

Antti Karttunen, Dec 16 2018

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A001359, A006512, A209328, A322356.

Programs

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 *)

PARI

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

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