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A071626 Number of distinct exponents in the prime factorization of n!. 10

%I

%S 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,

%T 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,

%U 10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11

%N Number of distinct exponents in the prime factorization of n!.

%C 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

%H P. Erdős, <a href="https://users.renyi.hu/~p_erdos/1982-08.pdf">Miscellaneous problems in number theory</a>, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congressus Numerantium 34 (1982), 25-45.

%F a(n) = A071625(n!) = A323023(n!,3). - _Gus Wiseman_, May 15 2019

%e n=7: 7! = 5040 = 2*2*2*2*3*3*5*7; three different exponents arise: 4, 2 and 1; a(7)=3.

%t 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}]

%t Table[Length[Union[Last/@If[n==1,{},FactorInteger[n!]]]],{n,30}] (* _Gus Wiseman_, May 15 2019 *)

%o (PARI) a(n) = #Set(factor(n!)[, 2]); \\ _Michel Marcus_, Sep 05 2017

%Y Cf. A051903, A051904, A071625.

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

%Y Cf. A325272, A325273, A325276, A325508.

%K nonn

%O 1,4

%A _Labos Elemer_, May 29 2002

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Last modified October 20 22:37 EDT 2019. Contains 328291 sequences. (Running on oeis4.)