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A344687 a(n) is the lowest nonnegative exponent k such that n!^k is the product of the divisors of n!.

%I A344687 #74 Jun 25 2022 21:55:08
%S A344687 0,1,2,4,8,15,30,48,80,135,270,396,792,1296,2016,2688,5376,7344,14688,
%T A344687 20520,30400,48000,96000,121440,170016,266112,338688,458640,917280,
%U A344687 1166400,2332800,2764800,3932160,6082560,8211456,9797760,19595520,30233088,42550272
%N A344687 a(n) is the lowest nonnegative exponent k such that n!^k is the product of the divisors of n!.
%C A344687 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 A344687 For prime p, d(p!) = 2*d((p-1)!), so a(p) = 2*a(p-1).
%F A344687 a(n) = d(n!)/2 = A000005(A000142(n))/2 = A027423(n)/2 for n > 1.
%F A344687 a(n) = A157672(n-1) + 1 for all n >= 2.
%e A344687 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 A344687 Join[{0},Table[DivisorSigma[0,n!]/2,{n,2,39}]] (* _Stefano Spezia_, Aug 18 2021 *)
%o A344687 (Python)
%o A344687 def a(n):
%o A344687     d = {}
%o A344687     for i in range(2, n+1):
%o A344687         tmp = i
%o A344687         j = 2
%o A344687         while(tmp != 1):
%o A344687             if(tmp % j == 0):
%o A344687                 d.setdefault(j, 0)
%o A344687                 tmp //= j
%o A344687                 d[j] += 1
%o A344687             else:
%o A344687                 j += 1
%o A344687     res = 1
%o A344687     for i in d.values():
%o A344687         res *= (i+1)
%o A344687     return res // 2
%o A344687 (Python)
%o A344687 from math import prod
%o A344687 from collections import Counter
%o A344687 from sympy import factorint
%o A344687 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 A344687 (PARI) a(n) = if (n==1, 0, numdiv(n!)/2); \\ _Michel Marcus_, Aug 18 2021
%Y A344687 Cf. A000005, A000142, A027423, A280420, A157672.
%K A344687 nonn
%O A344687 1,3
%A A344687 _Alex Sokolov_, Aug 17 2021