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

1,3

Comments

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).

Links

Table of n, a(n) for n=1..39.

Formula

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A176503 A262333 A293335 * A301480 A217777 A034338

Adjacent sequences: A344684 A344685 A344686 * A344688 A344689 A344690

Keyword

nonn

Author

Alex Sokolov, Aug 17 2021

Status

approved