A279952 - OEIS

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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

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OFFSET

1,15

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

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

EXAMPLE

a(15) = 2 because 15 has 4 divisors {1,3,5,15} among which 2 divisors {3,5} are primes with prime subscripts.

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 *)

PROG

(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

CROSSREFS

Cf. A001221, A006450, A257994.

Cf. A000720, A111406.

Sequence in context: A374060 A375149 A113706 * A054845 A317991 A359980

Adjacent sequences: A279949 A279950 A279951 * A279953 A279954 A279955

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Dec 23 2016

STATUS

approved