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

0,3

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.

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

Jon Maiga, Computer-generated formulas for A336415, Sequence Machine.

Formula

a(n) = A327527(n!).

Example

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

Mathematica

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

Prog

(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<<t - 1) ); res }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Jul 27 2020

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.

Sequence in context: A007981 A258847 A238627 * A241821 A280862 A370654

Adjacent sequences: A336412 A336413 A336414 * A336416 A336417 A336418

Keyword

nonn

Author

Gus Wiseman, Jul 22 2020

Extensions

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

Status

approved