A135291 Product of the nonzero exponents in the prime factorization of n!.

1, 1, 1, 1, 3, 3, 8, 8, 14, 28, 64, 64, 100, 100, 220, 396, 540, 540, 768, 768, 1152, 1944, 4104, 4104, 5280, 7920, 16560, 21528, 31200, 31200, 40768, 40768, 48608, 78120, 161280, 230400, 277440, 277440, 571200, 907200, 1108080, 1108080, 1440504, 1440504, 2019168

Offset

0,5

Comments

a(n) = A005361(n!). For n >= 2, a(n) = the number of positive divisors of n! which themselves are each divisible by every prime <= n. For p = any prime, a(p) = a(p-1). a(0)=a(1)=1 because the product of the exponents is over the empty set.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from G. C. Greubel)

Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Formula

a(n) = A000005(A049614(n)). - Ridouane Oudra, Sep 02 2019

a(n) = exp((n/log(n)) * (Sum_{k=0..M} e_k/log(n)^k) + O(n/log(n)^(M+2))) for any given integer M >= 0, where e_k = k! * Sum_{j=0..k} (1/j!) * Sum_{s>=1} (log(s+1)^j/(s+1))*log(1+1/s) are constants (e_0 = A085361) (De Koninck and Verreault, 2024, p. 54, Theorem 4.4). - Amiram Eldar, Dec 10 2024

Example

6! = 720 has a prime factorization of 2^4 * 3^2 * 5^1. So a(6) = 4*2*1 = 8.
Also, 720 is divisible by a(6)=8 positive divisors which themselves are each divisible by every prime <= 6 (i.e., are each divisible by 2*3*5 = 30): 30, 60, 90, 120, 180, 240, 360, 720.

Maple

A005361 := proc(n) mul( op(2, i), i=ifactors(n)[2]) ; end: A135291 := proc(n) A005361(n!) ; end: seq(A135291(n), n=0..50) ; # R. J. Mathar, Dec 12 2007
# Alternative:
b:= proc(n) option remember; `if`(n<1, 1,
      b(n-1)+add(i[2]*x^i[1], i=ifactors(n)[2]))
    end:
a:= n-> mul(i, i=coeffs(b(n))):
seq(a(n), n=0..44);  # Alois P. Heinz, Jun 02 2025

Mathematica

Table[Product[FactorInteger[n! ][[i, 2]], {i, 1, Length[FactorInteger[n! ]]}], {n, 0, 50}] (* Stefan Steinerberger, Dec 05 2007 *)
Table[Times@@Transpose[FactorInteger[n!]][[2]], {n, 0, 50}] (* Harvey P. Dale, Aug 16 2011 *)

Prog

(PARI) valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2, n\2, s*=valp(n, p)); s \\ Charles R Greathouse IV, Oct 09 2016
(Python)
from math import prod
from collections import Counter
from sympy import factorint
def A135291(n): return prod(sum((Counter(factorint(i)) for i in range(2, n+1)), start=Counter()).values()) # Chai Wah Wu, Jun 02 2025

Crossrefs

Cf. A005361, A000005, A049614, A085361.

Sequence in context: A093366 A204136 A168283 * A058617 A205977 A363725

Adjacent sequences: A135288 A135289 A135290 * A135292 A135293 A135294

Keyword

nonn

Author

Leroy Quet, Dec 03 2007

Extensions

More terms from Stefan Steinerberger and R. J. Mathar, Dec 05 2007

Status

approved