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%I #48 Jun 11 2023 12:19:19
%S 1,7,8,13,14,19,20,23,24,25,26,31,32,33,34,37,38,43,44,47,48,49,50,53,
%T 54,55,56,61,62,63,64,67,68,73,74,75,76,79,80,83,84,85,86,89,90,91,92,
%U 93,94,97,98,103,104,109,110,113,114,115,116,117,118,119,120
%N Numbers r such that the r-th tetrahedral number A000292(r) divides r!.
%C Conjecture: for every odd integer r > 1, the following statements are equivalent: a) r is a term of this sequence, b) r + 1 is a term of this sequence, c) r + 2 is composite.
%H David A. Corneth, <a href="/A308833/b308833.txt">Table of n, a(n) for n = 1..10000</a>
%e The 7th tetrahedral number is 84, and 84*60 = 5040 = 7!.
%p q := n -> (irem(n!, n*(n+1)*(n+2)/6) = 0):
%p select(q, [$1..120])[];
%t Select[Range@ 120, Mod[#!, Pochhammer[#, 3]/6] == 0 &] (* _Michael De Vlieger_, Jul 08 2019 *)
%o (PARI) isok(k) = !(k! % (k*(k+1)*(k+2)/6)); \\ _Michel Marcus_, Jun 28 2019
%o (PARI) is(n) = { my(f = factor(binomial(n + 2, 3))); forstep(i = #f~, 1, -1, if(val(n, f[i, 1]) - f[i, 2] < 0, return(0) ) ); 1 }
%o val(n, p) = my(r=0); while(n, r+=n\=p);r \\ _David A. Corneth_, Mar 22 2021
%Y Cf. A000142 (factorial numbers), A000292 (tetrahedral numbers).
%Y Cf. A007921 (numbers which are not difference of two primes), A153238.
%K nonn,easy
%O 1,2
%A _Lorenzo Sauras Altuzarra_, Jun 28 2019
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