A095691 - OEIS

%I #20 Nov 02 2023 10:28:06

%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,4,1,1,

%T 1,4,1,1,1,3,1,1,1,2,2,1,1,3,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,4,1,1,1,2,

%U 1,1,1,6,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1,3,1,2,1,2,1,1,1,4,1,2,2,4,1,1,1,3,1

%N Multiplicative with a(p^e) = A000720(e)+1.

%C The number of divisors of n that are terms of A056166. - _Amiram Eldar_, Oct 31 2023

%H Antti Karttunen, <a href="/A095691/b095691.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 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{q prime} 1/p^q) = Sum_{n>=1} 1/A056166(n) = 1.80728269690724154161... . - _Amiram Eldar_, Oct 31 2023

%t Array[Times @@ Map[PrimePi@# + 1 &, FactorInteger[#][[All, -1]] ] &, 120] (* _Michael De Vlieger_, Jul 19 2017 *)

%o (PARI) A095691(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= (1+primepi(f[k, 2])); ); m; } \\ _Antti Karttunen_, Jul 19 2017

%o (Python)

%o from sympy import factorint, primepi, prod

%o def a(n): return 1 if n==1 else prod(primepi(e) + 1 for e in factorint(n).values())

%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Jul 19 2017

%Y Cf. A000720, A056166, A095683.

%K mult,nonn,easy

%O 1,4

%A _Vladeta Jovovic_, Jul 06 2004