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

%I A135291 #38 Jun 02 2025 14:40:01
%S A135291 1,1,1,1,3,3,8,8,14,28,64,64,100,100,220,396,540,540,768,768,1152,
%T A135291 1944,4104,4104,5280,7920,16560,21528,31200,31200,40768,40768,48608,
%U A135291 78120,161280,230400,277440,277440,571200,907200,1108080,1108080,1440504,1440504,2019168
%N A135291 Product of the nonzero exponents in the prime factorization of n!.
%C A135291 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.
%H A135291 Alois P. Heinz, <a href="/A135291/b135291.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from G. C. Greubel)
%H A135291 Jean-Marie De Koninck and William Verreault, <a href="https://doi.org/10.2298/PIM2429045D">Arithmetic functions at factorial arguments</a>, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.
%F A135291 a(n) = A000005(A049614(n)). - _Ridouane Oudra_, Sep 02 2019
%F A135291 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
%e A135291 6! = 720 has a prime factorization of 2^4 * 3^2 * 5^1. So a(6) = 4*2*1 = 8.
%e A135291 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.
%p A135291 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
%p A135291 # second Maple program:
%p A135291 b:= proc(n) option remember; `if`(n<1, 1,
%p A135291       b(n-1)+add(i[2]*x^i[1], i=ifactors(n)[2]))
%p A135291     end:
%p A135291 a:= n-> mul(i, i=coeffs(b(n))):
%p A135291 seq(a(n), n=0..44);  # _Alois P. Heinz_, Jun 02 2025
%t A135291 Table[Product[FactorInteger[n! ][[i, 2]], {i, 1, Length[FactorInteger[n! ]]}], {n, 0, 50}] (* _Stefan Steinerberger_, Dec 05 2007 *)
%t A135291 Table[Times@@Transpose[FactorInteger[n!]][[2]],{n,0,50}] (* _Harvey P. Dale_, Aug 16 2011 *)
%o A135291 (PARI) valp(n,p)=my(s); while(n\=p, s+=n); s
%o A135291 a(n)=my(s=1); forprime(p=2,n\2, s*=valp(n,p)); s \\ _Charles R Greathouse IV_, Oct 09 2016
%o A135291 (Python)
%o A135291 from math import prod
%o A135291 from collections import Counter
%o A135291 from sympy import factorint
%o A135291 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
%Y A135291 Cf. A005361, A000005, A049614, A085361.
%K A135291 nonn
%O A135291 0,5
%A A135291 _Leroy Quet_, Dec 03 2007
%E A135291 More terms from _Stefan Steinerberger_ and _R. J. Mathar_, Dec 05 2007