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%I #13 Oct 20 2023 17:00:38
%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 V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].
%H Amiram Eldar, <a href="/A177329/b177329.txt">Table of n, a(n) for n = 2..1000</a>
%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 A000120(e_i), where n! = product 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 b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], _?(# == 1 &)]) // Flatten; a[n_] := Module[{np = PrimePi[n]}, v = Table[0, {np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j, 2, n}], {k, 1, np}]; Length[(b /@ v) // Flatten]]; Array[a, 77, 2] (* _Amiram Eldar_, Sep 17 2019 *)
%Y Cf. A050376, A176525, A001358, A176472, A176509, A064380, A050292.
%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
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