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 A177329 #35 Apr 14 2025 16:17:46
%S A177329 1,2,3,4,3,4,6,6,4,5,7,8,9,10,11,12,8,9,9,11,12,13,13,14,15,16,14,15,
%T A177329 16,17,19,21,17,16,15,16,17,18,19,20,22,23,21,21,21,22,23,22,23,25,22,
%U A177329 23,22,24,26,28,28,29,27,28,29,30,32,34,30,31,31,28,27,28,29,30,31,33,31,31,30
%N A177329 Number of factors in the representation of n! as a product of distinct terms of A050376.
%D A177329 Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].
%H A177329 Chai Wah Wu, <a href="/A177329/b177329.txt">Table of n, a(n) for n = 2..10000</a> (terms 2..1000 from Amiram Eldar)
%H A177329 Simon Litsyn and Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%F A177329 a(n) = Sum_{i} A000120(e_i), where n! = Product_{i} p_i^e_i is the prime factorization of n!.
%F A177329 a(n) = A064547(n!). - _R. J. Mathar_, May 28 2010
%p A177329 read("transforms") ; A064547 := proc(n) f := ifactors(n)[2] ; a := 0 ; for p in f do a := a+wt(op(2,p)) ; end do: a ; end proc:
%p A177329 A177329 := proc(n) A064547(n!) ; end proc: seq(A177329(n),n=2..80) ; # _R. J. Mathar_, May 28 2010
%t A177329 f[p_, e_] := DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n!]; Array[a, 100, 2] (* _Amiram Eldar_, Aug 24 2024 *)
%o A177329 (Python)
%o A177329 from collections import Counter
%o A177329 from sympy import factorint
%o A177329 def A177329(n): return sum(map(int.bit_count,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())) # _Chai Wah Wu_, Jul 18 2024
%o A177329 (PARI) a(n) = vecsum(apply(x -> hammingweight(x), factor(n!)[,2])); \\ _Amiram Eldar_, Aug 24 2024
%Y A177329 Cf. A000120, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A176525.
%K A177329 nonn
%O A177329 2,2
%A A177329 _Vladimir Shevelev_, May 06 2010
%E A177329 a(20)=10 inserted by _Vladimir Shevelev_, May 08 2010
%E A177329 Terms from a(14) onwards replaced according to the formula - _R. J. Mathar_, May 28 2010