This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384815 #15 Jul 03 2025 08:40:19
%S A384815 0,1,1,8,1,2,1,27,8,2,1,9,1,2,2,64,1,9,1,9,2,2,1,28,8,2,27,9,1,3,1,
%T A384815 125,2,2,2,16,1,2,2,28,1,3,1,9,9,2,1,65,8,9,2,9,1,28,2,28,2,2,1,10,1,
%U A384815 2,9,216,2,3,1,9,2,3,1,35,1,2,9,9,2,3,1,65,64,2,1,10,2,2,2,28,1,10
%N A384815 Sum of the cubes of the exponents in the prime factorization of n.
%H A384815 Amiram Eldar, <a href="/A384815/b384815.txt">Table of n, a(n) for n = 1..10000</a>
%H A384815 R. L. Duncan, <a href="https://www.jstor.org/stable/2312731">A class of additive arithmetical functions</a>, The American Mathematical Monthly, Vol. 69, No. 1 (1962), pp. 34-36.
%H A384815 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A384815 If n = Product (p_j^k_j) then a(n) = Sum (k_j^3).
%F A384815 From _Amiram Eldar_, Jul 03 2025: (Start)
%F A384815 Additive with a(p^e) = e^3.
%F A384815 Sum_{k=1..n} a(k) ~ n * log(log(n)) + B_3 * n + O(n/log(n)), where B_3 = gamma + Sum_{p prime} ((1-1/p)*Sum_{m>=1} m^3/p^m + log(1-1/p)) = 16.17021843694072992072..., and gamma is Euler's constant (A001620) (Duncan, 1962). (End)
%t A384815 Join[{0}, Table[Plus @@ (#[[2]]^3 & /@ FactorInteger[n]), {n, 2, 90}]]
%o A384815 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]^3); \\ _Michel Marcus_, Jun 10 2025
%Y A384815 Cf. A001222, A001620, A005064, A090885, A360970.
%K A384815 nonn,easy
%O A384815 1,4
%A A384815 _Ilya Gutkovskiy_, Jun 10 2025