cp's OEIS Frontend

A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.

%I A136565 #27 Jun 30 2025 04:25:24
%S A136565 0,1,1,2,1,1,1,3,2,1,1,3,1,1,1,4,1,3,1,3,1,1,1,4,2,1,3,3,1,1,1,5,1,1,
%T A136565 1,2,1,1,1,4,1,1,1,3,3,1,1,5,2,3,1,3,1,4,1,4,1,1,1,3,1,1,3,6,1,1,1,3,
%U A136565 1,1,1,5,1,1,3,3,1,1,1,5,4,1,1,3,1,1,1,4,1,3,1,3,1,1,1,6,1,3,3,2,1,1,1,4,1
%N A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.
%C A136565 The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 192, 2089, 21405, 215730, 2162136, 21636277, 216410510, 2164253043, 21642998932, ... . Apparently, the asymptotic mean of this sequence is 2.164... . - _Amiram Eldar_, Jun 30 2025
%H A136565 Antti Karttunen, <a href="/A136565/b136565.txt">Table of n, a(n) for n = 1..65537</a> (terms 1..1000 from Diana Mecum)
%H A136565 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A136565 a(n) = A088529(n) = A181591(n) for n: 2 <= n < 24. - _Reinhard Zumkeller_, Nov 01 2010
%F A136565 a(n) = A066328(A181819(n)). - _Antti Karttunen_, Sep 06 2018
%e A136565 120 = 2^3 * 3^1 * 5^1. The exponents of the prime factorization are therefore 3,1,1. The distinct values which equal these exponents are 1 and 3. So a(120) = 1+3 = 4.
%t A136565 Join[{0},Table[Total[Union[Transpose[FactorInteger[n]][[2]]]],{n,2,110}]] (* _Harvey P. Dale_, Jun 23 2013 *)
%o A136565 (PARI) A136565(n) = vecsum(apply(primepi,factor(factorback(apply(e->prime(e),(factor(n)[,2]))))[,1])); \\ _Antti Karttunen_, Sep 06 2018
%Y A136565 Cf. A066328, A071625, A088529, A136566, A136568, A181591, A181819.
%K A136565 nonn
%O A136565 1,4
%A A136565 _Leroy Quet_, Jan 07 2008
%E A136565 More terms from _Diana L. Mecum_, Jul 17 2008