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).
FORMULA
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
KEYWORD
nonn
AUTHOR
Alex Sokolov, Aug 17 2021
STATUS
approved