A177329 - OEIS

%I #33 Aug 24 2024 06:00:15

%S 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 16,17,19,21,17,16,15,16,17,18,19,20,22,23,21,21,21,22,23,22,23,25,22,

%U 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 Number of factors in the representation of n! as a product of distinct terms of A050376.

%D 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 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 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 a(n) = Sum_{i} A000120(e_i), where n! = Product_{i} p_i^e_i is the prime factorization of n!.

%F a(n) = A064547(n!). - _R. J. Mathar_, May 28 2010

%p 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 := proc(n) A064547(n!) ; end proc: seq(A177329(n),n=2..80) ; # _R. J. Mathar_, May 28 2010

%t 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 (Python)

%o from collections import Counter

%o from sympy import factorint

%o 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 (PARI) a(n) = vecsum(apply(x -> hammingweight(x), factor(n!)[,2])); \\ _Amiram Eldar_, Aug 24 2024

%Y Cf. A000120, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A176525.

%K nonn

%O 2,2

%A _Vladimir Shevelev_, May 06 2010

%E I inserted one omitted term: a(20)=10. _Vladimir Shevelev_, May 08 2010

%E Terms from a(14) onwards replaced according to the formula - _R. J. Mathar_, May 28 2010