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

%I A352447 #22 Mar 25 2022 09:26:21
%S A352447 1,2,7,9,10,11,13,15,16,17,19,21,22,23,25,26,27,28,29,31,33,34,35,36,
%T A352447 37,39,40,41,43,45,46,47,49,50,51,52,53,55,56,57,58,59,61,63,64,65,66,
%U A352447 67,69,70,71,73,75,76,77,78,79,81,82,83,85,86,87,88,89,91,92,93,94,95
%N A352447 Numbers k such that BarnesG(k) is divisible by Gamma(k).
%C A352447 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
%H A352447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%H A352447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisible.html">Divisible</a>.
%H A352447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a>.
%H A352447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Superfactorial.html">Superfactorial</a>.
%F A352447 Conjecture: a(n) = A079696(n-1), n>1. - _R. J. Mathar_, Mar 20 2022
%e A352447 BarnesG(7) = 34560, Gamma(7) = 720, 34560 is divisible by 720, so 7 is in this sequence.
%t A352447 Table[If[Divisible[BarnesG[k], Gamma[k]], k, Nothing], {k, 115}]
%o A352447 (Python)
%o A352447 from itertools import count, islice
%o A352447 from collections import Counter
%o A352447 from sympy import factorint
%o A352447 def A352447_gen(): # generator of terms
%o A352447     yield 1
%o A352447     a = Counter()
%o A352447     for k in count(2):
%o A352447         b = Counter(factorint(k-1))
%o A352447         if all(b[p] <= a[p] for p in b):
%o A352447             yield k
%o A352447         a += b
%o A352447 A352447_list = list(islice(A352447_gen(),100)) # _Chai Wah Wu_, Mar 17 2022
%Y A352447 Cf. A000142, A000178, A079696.
%K A352447 nonn
%O A352447 1,2
%A A352447 _Vladimir Reshetnikov_, Mar 16 2022