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

1,2

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A343298 A061905 A308887 * A025156 A228210 A045765

Adjacent sequences: A308830 A308831 A308832 * A308834 A308835 A308836

Keyword

nonn,easy

Author

Lorenzo Sauras Altuzarra, Jun 28 2019

Status

approved