|
%I #20 Sep 29 2023 04:02:31
%S 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,
%T 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,
%U 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
%N Number of exponents in prime factorization of n which are primes.
%C First occurrence of k: 1,4,36,900,44100 (A061742). - _Robert G. Wilson v_, Jan 25 2005
%H Antti Karttunen, <a href="/A101436/b101436.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F Additive with a(p^e) = A010051(e). - _Antti Karttunen_, Jul 19 2017
%F 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
%e 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.
%t f[n_] := Length[ Select[ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]], PrimeQ[ # ] &]]; Table[ f[n], {n, 105}] (* _Robert G. Wilson v_, Jan 25 2005 *)
%t Table[Count[Transpose[FactorInteger[n]][[2]],_?PrimeQ],{n,120}] (* _Harvey P. Dale_, Mar 21 2016 *)
%o (PARI) A101436(n) = vecsum(apply(e -> isprime(e), factorint(n)[, 2])); \\ _Antti Karttunen_, Jul 19 2017
%Y Cf. A010051, A061742, A123391, A125029, A125070, A125072.
%K nonn,easy
%O 1,36
%A _Leroy Quet_, Jan 18 2005
%E More terms from _Robert G. Wilson v_, Jan 25 2005
|
