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 A081402 #22 Apr 05 2025 06:33:33
%S A081402 0,2,5,11,16,30,37,149,221,369,380,1310,1323,2389,2975,33695,33712,
%T A081402 72312,72331,269439,282855,545109,545132,4254514,4269514,8463974,
%U A081402 9999248,35167130,35167159,71972737,71972768,2152347552,2161914700
%N A081402 a(n) = A008475(n!).
%H A081402 Jean-Marie De Koninck and William Verreault, <a href="https://doi.org/10.2298/PIM2429045D">Arithmetic functions at factorial arguments</a>, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.
%F A081402 From _Amiram Eldar_, Dec 10 2024: (Start)
%F A081402 a(n) = 2^(n-s_2(n)) + O(sqrt(3)^n), where s_2(n) = A000120(n).
%F A081402 Sum_{k=1..n} a(k) = 2^(n+O(log(n))).
%F A081402 Both formulas from De Koninck and Verreault (2024, pp. 51-52, eq. (3.10) and (3.16)). (End)
%e A081402 a(1) = 0 since 1! = 1 has no prime factor.
%e A081402 a(8) = 2^7 + 3^2 + 5 + 7 = 149 since 8! = 2^7*3^2*5*7.
%t A081402 ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}]; supo[x_] := Apply[Plus, ba[x]^ep[x]]; Table[supo[w], {w, 1, 25}]
%o A081402 (PARI) a(n) = my(f=factor(n!)); sum(k=1, #f~, f[k,1]^f[k,2]); \\ _Michel Marcus_, Jul 09 2018
%Y A081402 Cf. A008475, A000120, A000142, A081403, A081404.
%K A081402 nonn
%O A081402 1,2
%A A081402 _Labos Elemer_, Mar 31 2003