login
A279952
Number of primes with prime subscripts dividing n.
7
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
1,15
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Dec 23 2016
STATUS
approved