OFFSET
1,2
COMMENTS
These are k such that G(k)/Gamma(k) = 1!*2!*3!*...*(k-2)!/(k-1)! = 1!*2!*3!*...*(k-3)!/(k-1) are integer. Let k=1+c, so require 1!*2!*3!*...*(c-2)!/c to be integer. If c is composite, take any factorization of c=c_1*c_2 with 2<=c_1<=c_2<=c/2; then matching factors exist in the product 1!*2!*3!*...*(c-2)! that cancel this factor [either c_1! and c_2! if c_1 <> c_2, or c_1! and (c_1+1)! if c_1=c_2 and c-2 >= c_1+1]. If c is prime, none of the 1!*2!*..(c-2)! contains a factor matching that prime. So this shows that the sequence is (apart from offset and at c=4) the same as A079696. - R. J. Mathar, Mar 25 2022
LINKS
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Divisible.
Eric Weisstein's World of Mathematics, Gamma Function.
Eric Weisstein's World of Mathematics, Superfactorial.
FORMULA
Conjecture: a(n) = A079696(n-1), n>1. - R. J. Mathar, Mar 20 2022
EXAMPLE
BarnesG(7) = 34560, Gamma(7) = 720, 34560 is divisible by 720, so 7 is in this sequence.
MATHEMATICA
Table[If[Divisible[BarnesG[k], Gamma[k]], k, Nothing], {k, 115}]
PROG
(Python)
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A352447_gen(): # generator of terms
yield 1
a = Counter()
for k in count(2):
b = Counter(factorint(k-1))
if all(b[p] <= a[p] for p in b):
yield k
a += b
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Mar 16 2022
STATUS
approved