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
Keywords
Examples
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.
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.
Programs
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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 # second Maple program: 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
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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 *) -
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
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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
Formula
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
Extensions
More terms from Stefan Steinerberger and R. J. Mathar, Dec 05 2007
Comments