A279952 Number of primes with prime subscripts dividing n.
0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 2, 0, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2
Offset: 1
Examples
a(15) = 2 because 15 has 4 divisors {1,3,5,15} among which 2 divisors {3,5} are primes with prime subscripts.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
Programs
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Maple
with(numtheory): seq(add(pi(pi(d))-pi(pi(d-1)), d in divisors(n)), n=1..80); # Ridouane Oudra, Sep 12 2023
-
Mathematica
Rest[nmax = 120; CoefficientList[Series[Sum[x^Prime[Prime[k]]/(1 - x^Prime[Prime[k]]), {k, 1, nmax}], {x, 0, nmax}], x]] f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 03 2023 *) -
PARI
my(x='x+O('x^120)); concat([0, 0], Vec(sum(k=1, 120, x^prime(prime(k))/(1 - x^prime(prime(k)))))) \\ Indranil Ghosh, May 23 2017
Formula
G.f.: Sum_{k>=1} x^prime(prime(k))/(1 - x^prime(prime(k))).
a(n) = Sum_{d|n} A111406(d-1). - Ridouane Oudra, Sep 12 2023
Additive with a(p^e) = 1 if primepi(p) is prime, and 0 otherwise. - Amiram Eldar, Nov 03 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A006450(n) = 1.04... (see A006450 for a better estimate of this constant). - Amiram Eldar, Jan 01 2024