%I #17 Aug 08 2024 06:45:38

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

%H Amiram Eldar, <a href="/A133924/b133924.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A136567(n!). - _Amiram Eldar_, Aug 08 2024

%e 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 := 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 ne1[n_]:=Count[Tally[Transpose[FactorInteger[n!]][[2]]],_?(Last[#] == 1&)]; Join[{0,0},Array[ne1,110,2]] (* _Harvey P. Dale_, Aug 21 2011 *)

%Y Cf. A000142, A071626, A136567.

%K nonn,changed

%O 0,5

%A _Leroy Quet_, Jan 07 2008

%E More terms from _R. J. Mathar_, Jan 30 2008