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A071626 Number of distinct exponents in the prime factorization of n!. 10
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..92.

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.

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.

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.

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

Cf. A325272, A325273, A325276, A325508.

Sequence in context: A268708 A061555 A146323 * A182008 A106457 A103128

Adjacent sequences:  A071623 A071624 A071625 * A071627 A071628 A071629

KEYWORD

nonn

AUTHOR

Labos Elemer, May 29 2002

STATUS

approved

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Last modified August 18 07:11 EDT 2019. Contains 326072 sequences. (Running on oeis4.)