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

1,4

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

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.

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A051903, A051904, A071625, A240751.

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

Cf. A325272, A325273, A325276, A325508.

Sequence in context: A268708 A061555 A146323 * A182008 A375024 A106457

Adjacent sequences: A071623 A071624 A071625 * A071627 A071628 A071629

Keyword

nonn,easy

Author

Labos Elemer, May 29 2002

Status

approved