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

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, 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, 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

Offset: 0

Author

Leroy Quet, Jan 07 2008

Keywords

• nonn

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000142, A071626, A136567.

Programs

Maple

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

Mathematica

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

Formula

a(n) = A136567(n!). - Amiram Eldar, Aug 08 2024

Extensions

More terms from R. J. Mathar, Jan 30 2008