A308833 Numbers r such that the r-th tetrahedral number A000292(r) divides r!.

1, 7, 8, 13, 14, 19, 20, 23, 24, 25, 26, 31, 32, 33, 34, 37, 38, 43, 44, 47, 48, 49, 50, 53, 54, 55, 56, 61, 62, 63, 64, 67, 68, 73, 74, 75, 76, 79, 80, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 97, 98, 103, 104, 109, 110, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Author

Lorenzo Sauras Altuzarra, Jun 28 2019

Keywords

• nonn
• easy

Comments

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.

Examples

			The 7th tetrahedral number is 84, and 84*60 = 5040 = 7!.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000142 (factorial numbers), A000292 (tetrahedral numbers).

Cf. A007921 (numbers which are not difference of two primes), A153238.

Programs

Maple

q := n -> (irem(n!, n*(n+1)*(n+2)/6) = 0):
select(q, [$1..120])[];

Mathematica

Select[Range@ 120, Mod[#!, Pochhammer[#, 3]/6] == 0 &] (* Michael De Vlieger, Jul 08 2019 *)

PARI

isok(k) = !(k! % (k*(k+1)*(k+2)/6)); \\ Michel Marcus, Jun 28 2019

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 }
val(n, p) = my(r=0); while(n, r+=n\=p);r \\ David A. Corneth, Mar 22 2021