A280420 -id:A280420 - cp's OEIS frontend

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

0, 1, 2, 4, 8, 15, 30, 48, 80, 135, 270, 396, 792, 1296, 2016, 2688, 5376, 7344, 14688, 20520, 30400, 48000, 96000, 121440, 170016, 266112, 338688, 458640, 917280, 1166400, 2332800, 2764800, 3932160, 6082560, 8211456, 9797760, 19595520, 30233088, 42550272

Offset: 1

Alex Sokolov, Aug 17 2021

nonn

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.

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

			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.

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

Join[{0},Table[DivisorSigma[0,n!]/2,{n,2,39}]] (* Stefano Spezia, Aug 18 2021 *)

a(n) = if (n==1, 0, numdiv(n!)/2); \\ Michel Marcus, Aug 18 2021

def a(n):
    d = {}
    for i in range(2, n+1):
        tmp = i
        j = 2
        while(tmp != 1):
            if(tmp % j == 0):
                d.setdefault(j, 0)
                tmp //= j
                d[j] += 1
            else:
                j += 1
    res = 1
    for i in d.values():
        res *= (i+1)
    return res // 2

from math import prod
from collections import Counter
from sympy import factorint
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

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

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

cp's OEIS Frontend

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

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