

A308833


Numbers r such that the rth tetrahedral number A000292(r) divides r!.


1



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
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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



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 }


CROSSREFS

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


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



