This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A071626 #47 Apr 10 2024 11:11:39
%S A071626 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 A071626 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 A071626 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 A071626 Number of distinct exponents in the prime factorization of n!.
%C A071626 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
%C A071626 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
%H A071626 David A. Corneth, <a href="/A071626/b071626.txt">Table of n, a(n) for n = 1..10000</a>
%H A071626 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.
%H A071626 Randell Heyman and Md Rahil Miraj, <a href="https://arxiv.org/abs/2309.16072">On some floor function sets</a>, arXiv:2309.16072 [math.NT], 2023-2024.
%F A071626 a(n) = A071625(n!) = A323023(n!,3). - _Gus Wiseman_, May 15 2019
%e A071626 n=7: 7! = 5040 = 2*2*2*2*3*3*5*7; three different exponents arise: 4, 2 and 1; a(7)=3.
%e A071626 n=7: { floor(7/p) : p <= 7, p prime } = {3,2,1}. So, its cardinality is 3. - _Md Rahil Miraj_, Apr 05 2024
%t A071626 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 A071626 Table[Length[Union[Last/@If[n==1,{},FactorInteger[n!]]]],{n,30}] (* _Gus Wiseman_, May 15 2019 *)
%o A071626 (PARI) a(n) = #Set(factor(n!)[, 2]); \\ _Michel Marcus_, Sep 05 2017
%Y A071626 Cf. A051903, A051904, A071625, A240751.
%Y A071626 Cf. A000142, A001221, A001222, A011371, A022559, A076934, A115627, A135291.
%Y A071626 Cf. A325272, A325273, A325276, A325508.
%K A071626 nonn,easy
%O A071626 1,4
%A A071626 _Labos Elemer_, May 29 2002