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A133924 a(n) = number of exponents occurring only once each in the prime factorization of n!.

%I A133924 #17 Aug 08 2024 06:45:38
%S A133924 0,0,1,0,2,1,3,2,2,2,4,3,3,3,2,4,4,4,4,4,4,4,5,5,5,5,4,4,4,4,4,4,4,6,
%T A133924 5,5,5,5,5,4,4,4,4,4,6,6,6,6,6,6,6,5,5,5,5,7,7,7,6,6,6,6,6,6,6,4,6,6,
%U A133924 8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,9,9,8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,9,9,9
%N A133924 a(n) = number of exponents occurring only once each in the prime factorization of n!.
%H A133924 Amiram Eldar, <a href="/A133924/b133924.txt">Table of n, a(n) for n = 0..10000</a>
%F A133924 a(n) = A136567(n!). - _Amiram Eldar_, Aug 08 2024
%e A133924 14! is factored into primes as 2^11 * 3^5 * 5^2 * 7^2 * 11^1 * 13^1. The exponent 1 and 2 each occur more than once. So the exponents occurring only once each are 5 and 11. Therefore a(14) = 2.
%p A133924 A133924 := proc(n) local ifs,a,i ; if n <= 1 then RETURN(0) ; else ifs := ifactors(n!)[2] ; ifs := sort([seq(op(2,i),i=ifs)]) ; a :=0 ; for i from 1 to nops(ifs) do if i = 1 or op(i,ifs) <> op(i-1,ifs) then if i=nops(ifs) or op(i,ifs) <> op(i+1,ifs) then a := a+1 ; fi ; fi ; od: RETURN(a) ; fi ; end: seq(A133924(n),n=0..120) ; # _R. J. Mathar_, Jan 30 2008
%t A133924 ne1[n_]:=Count[Tally[Transpose[FactorInteger[n!]][[2]]],_?(Last[#] == 1&)]; Join[{0,0},Array[ne1,110,2]] (* _Harvey P. Dale_, Aug 21 2011 *)
%Y A133924 Cf. A000142, A071626, A136567.
%K A133924 nonn
%O A133924 0,5
%A A133924 _Leroy Quet_, Jan 07 2008
%E A133924 More terms from _R. J. Mathar_, Jan 30 2008