A352447 Numbers k such that BarnesG(k) is divisible by Gamma(k).

1, 2, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Author

Vladimir Reshetnikov, Mar 16 2022

Keywords

• nonn

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

Examples

			BarnesG(7) = 34560, Gamma(7) = 720, 34560 is divisible by 720, so 7 is in this sequence.
		

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.

Crossrefs

Cf. A000142, A000178, A079696.

Programs

Mathematica

Table[If[Divisible[BarnesG[k], Gamma[k]], k, Nothing], {k, 115}]

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
A352447_list = list(islice(A352447_gen(),100)) # Chai Wah Wu, Mar 17 2022

Formula

Conjecture: a(n) = A079696(n-1), n>1. - R. J. Mathar, Mar 20 2022