A101436 Number of exponents in prime factorization of n which are primes.

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

Offset

1,36

Comments

First occurrence of k: 1,4,36,900,44100 (A061742). - Robert G. Wilson v, Jan 25 2005

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

Additive with a(p^e) = A010051(e). - Antti Karttunen, Jul 19 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (P(p)-P(p+1)) = 0.39847584805803104040..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 29 2023

Example

36 = 2^2 *3^2. Since 2 is a prime and occurs twice as an exponent in the prime factorization of 36, a(36) = 2.

Mathematica

f[n_] := Length[ Select[ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]], PrimeQ[ # ] &]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jan 25 2005 *)
Table[Count[Transpose[FactorInteger[n]][[2]], _?PrimeQ], {n, 120}] (* Harvey P. Dale, Mar 21 2016 *)

Prog

(PARI) A101436(n) = vecsum(apply(e -> isprime(e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 19 2017

Crossrefs

Cf. A010051, A061742, A123391, A125029, A125070, A125072.

Sequence in context: A160383 A330023 A328891 * A366247 A369165 A056170

Adjacent sequences: A101433 A101434 A101435 * A101437 A101438 A101439

Keyword

nonn,easy

Author

Leroy Quet, Jan 18 2005

Extensions

More terms from Robert G. Wilson v, Jan 25 2005

Status

approved