A336415 Number of divisors of n! with equal prime multiplicities.

1, 1, 2, 4, 6, 10, 13, 21, 24, 28, 33, 49, 53, 85, 94, 100, 104, 168, 173, 301, 307, 317, 334, 590, 595, 603, 636, 642, 652, 1164, 1171, 2195, 2200, 2218, 2283, 2295, 2301, 4349, 4478, 4512, 4519, 8615, 8626, 16818, 16836, 16844, 17101, 33485, 33491, 33507, 33516, 33582

Offset: 0

Author

Gus Wiseman, Jul 22 2020

Keywords

• nonn

Comments

A number k has "equal prime multiplicities" (or is "uniform") iff its prime signature is constant, meaning that k is a power of a squarefree number.

Examples

			The a(n) uniform divisors of n for n = 1, 2, 6, 8, 30, 36 are the columns:
  1  2  6  8  30  36
     1  3  6  15  30
        2  4  10  16
        1  3   8  15
           2   6  10
           1   5   9
               4   8
               3   6
               2   5
               1   4
                   3
                   2
                   1
In 20!, the multiplicity of the third prime (5) is 4 but the multiplicity of the fourth prime (7) is 2. Hence there are 2^3 - 1 = 3 divisors with all exponents 3 (we subtract |{1}| = 1 from that count as 1 has no exponent 3). - _David A. Corneth_, Jul 27 2020
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999
• Jon Maiga, Computer-generated formulas for A336415, Sequence Machine.

Crossrefs

The version for distinct prime multiplicities is A336414.

The version for nonprime perfect powers is A336416.

Uniform partitions are counted by A047966.

Uniform numbers are A072774, with nonprime terms A182853.

Numbers with distinct prime multiplicities are A130091.

Divisors with distinct prime multiplicities are counted by A181796.

Maximum divisor with distinct prime multiplicities is A327498.

Uniform divisors are counted by A327527.

Maximum uniform divisor is A336618.

1st differences are given by A048675.

Cf. A000005, A000961, A001222, A001597, A007916, A112798, A124010, A327499.

Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617.

Programs

Mathematica

Table[Length[Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

PARI

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); (#ex==0) || (vecmin(ex) == vecmax(ex))); \\ Michel Marcus, Jul 24 2020

PARI

a(n) = {if(n<2, return(1)); my(f = primes(primepi(n)), res = 1, t = #f); f = vector(#f, i, val(n, f[i])); for(i = 1, f[1], while(f[t] < i, t--; ); res+=(1<David A. Corneth, Jul 27 2020

Formula

a(n) = A327527(n!).

Extensions

Terms a(31) and onwards from David A. Corneth, Jul 27 2020