A344687 - OEIS

%I #74 Jun 25 2022 21:55:08

%S 0,1,2,4,8,15,30,48,80,135,270,396,792,1296,2016,2688,5376,7344,14688,

%T 20520,30400,48000,96000,121440,170016,266112,338688,458640,917280,

%U 1166400,2332800,2764800,3932160,6082560,8211456,9797760,19595520,30233088,42550272

%N a(n) is the lowest nonnegative exponent k such that n!^k is the product of the divisors of n!.

%C This sequence is a subsequence of A001222, because the product of divisors of n! is n^(d(n)/2) (where d(n) is the number of divisors of n), so a(n) = d(n!)/2.

%C For prime p, d(p!) = 2*d((p-1)!), so a(p) = 2*a(p-1).

%F a(n) = d(n!)/2 = A000005(A000142(n))/2 = A027423(n)/2 for n > 1.

%F a(n) = A157672(n-1) + 1 for all n >= 2.

%e For n = 4, n! = 24 = 2^3 * 3, which has (3+1)*(1+1) = 8 divisors: {1,2,3,4,6,8,12,24} whose product is 331776 = (24)^4 = (4!)^4. So a(4) = 4.

%t Join[{0},Table[DivisorSigma[0,n!]/2,{n,2,39}]] (* _Stefano Spezia_, Aug 18 2021 *)

%o (Python)

%o def a(n):

%o     d = {}

%o     for i in range(2, n+1):

%o         tmp = i

%o         j = 2

%o         while(tmp != 1):

%o             if(tmp % j == 0):

%o                 d.setdefault(j, 0)

%o                 tmp //= j

%o                 d[j] += 1

%o             else:

%o                 j += 1

%o     res = 1

%o     for i in d.values():

%o         res *= (i+1)

%o     return res // 2

%o (Python)

%o from math import prod

%o from collections import Counter

%o from sympy import factorint

%o def A344687(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())//2 # _Chai Wah Wu_, Jun 25 2022

%o (PARI) a(n) = if (n==1, 0, numdiv(n!)/2); \\ _Michel Marcus_, Aug 18 2021

%Y Cf. A000005, A000142, A027423, A280420, A157672.

%K nonn

%O 1,3

%A _Alex Sokolov_, Aug 17 2021