A156542 Number of distinct Sophie Germain prime factors of n.

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 2, 2, 0, 2, 0, 2, 1, 1, 1, 2, 2, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 0, 2, 1, 1, 1, 2, 0, 1, 2, 2, 0, 2, 0, 1, 2

Offset: 1

Author

Reinhard Zumkeller, Feb 10 2009

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A001221, A005384, A053176, A156541, A156543, A101264.

Programs

Magma

[0] cat [&+[#PrimesInInterval(2*p,2*p+1):p in PrimeDivisors(n)]:n in [2..100]]; // Marius A. Burtea, Aug 25 2019

Maple

with(numtheory): A101264:=p-> pi(2*p+1)-pi(2*p): seq(add(A101264(p), p in factorset(n)), n=1..100); # Ridouane Oudra, Aug 25 2019

Mathematica

Join[{0},Table[Count[FactorInteger[n][[All,1]],?(PrimeQ[2#+1]&)],{n,2,110}]] (* _Harvey P. Dale, Apr 05 2020 *)

Formula

a(n) <= A001221(n).

a(A156541(n)) = A001221(A156541(n)); a(A156543(n)) = 0.

a(A005384(n)) = 1; a(A053176(n)) = 0.

a(n) = Sum_{p|n} A101264(p), where p is a prime. - Ridouane Oudra, Aug 25 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A005384(k) (see A005384 for an estimate of this sum). - Amiram Eldar, Jun 03 2024