A071626 Number of distinct exponents in the prime factorization of n!.

0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Labos Elemer, May 29 2002

Keywords

• nonn
• easy

Comments

Erdős proved that there exist two constants c1, c2 > 0 such that c1 (n / log(n))^(1/2) < a(n) < c2 (n / log(n))^(1/2). - Carlo Sanna, May 28 2019

R. Heyman and R. Miraj proved that the cardinality of the set { floor(n/p) : p <= n, p prime } is same as the number of distinct exponents in the prime factorization of n!. - Md Rahil Miraj, Apr 05 2024

Examples

			n=7: 7! = 5040 = 2*2*2*2*3*3*5*7; three different exponents arise: 4, 2 and 1; a(7)=3.
n=7: { floor(7/p) : p <= 7, p prime } = {3,2,1}. So, its cardinality is 3. - _Md Rahil Miraj_, Apr 05 2024
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• P. Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congressus Numerantium 34 (1982), 25-45.
• Randell Heyman and Md Rahil Miraj, On some floor function sets, arXiv:2309.16072 [math.NT], 2023-2024.

Crossrefs

Cf. A051903, A051904, A071625, A240751.

Cf. A000142, A001221, A001222, A011371, A022559, A076934, A115627, A135291.

Cf. A325272, A325273, A325276, A325508.

Programs

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] Table[Length[Union[ep[w! ]]], {w, 1, 100}]
Table[Length[Union[Last/@If[n==1,{},FactorInteger[n!]]]],{n,30}] (* Gus Wiseman, May 15 2019 *)

PARI

a(n) = #Set(factor(n!)[, 2]); \\ Michel Marcus, Sep 05 2017

Formula

a(n) = A071625(n!) = A323023(n!,3). - Gus Wiseman, May 15 2019