A342091 a(n) is the least number k such that k! has n distinct exponents in its prime factorization.

1, 2, 4, 6, 10, 15, 22, 33, 44, 55, 68, 85, 102, 119, 145, 174, 203, 232, 261, 296, 333, 370, 410, 451, 492, 533, 590, 656, 708, 767, 826, 885, 944, 1005, 1072, 1143, 1207, 1278, 1422, 1455, 1562, 1652, 1778, 1917, 2032, 2134, 2235, 2328, 2425, 2540, 2682, 2831, 2929

Offset

0,2

Comments

After n=0, first differs from A073818 at n = 27.

a(n) is the least k such that A071625(k!) = A071626(k) = n.

Is this sequence strictly increasing?

Links

Amiram Eldar, Table of n, a(n) for n = 0..2109 (terms below 10^7)

Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer., Vol. 34 (1982), pp. 25-45.

Example

a(1) = 2 since 2! = 2^1 is the least factorial with a single exponent (1) in its prime factorization.
a(2) = 4 since 4! = 24 = 2^3 * 3^1 is the least factorial with 2 distinct exponents (1 and 3) in its prime factorization.
a(3) = 6 since 6! = 720 = 2^4 * 3^2 * 5^1 is the least factorial with 3 distinct exponents (1, 2 and 4) in its prime factorization.

Mathematica

f[1] = 0; f[n_] := Length @ Union[FactorInteger[n!][[;; , 2]]]; seq[max_] := Module[{s = Table[0, {max}], n = 1, c = 0}, While[c < max, i = f[n] + 1; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50]

Crossrefs

Cf. A000142, A071625, A071626, A073818, A133924, A336616, A336619, A336867, A336868.

Sequence in context: A086182 A035294 A073818 * A239288 A306145 A143184

Adjacent sequences: A342088 A342089 A342090 * A342092 A342093 A342094

Keyword

nonn

Author

Amiram Eldar, Feb 27 2021

Status

approved