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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 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.)