A319788 Tetrahedral numbers divisible by a record number of smaller tetrahedral numbers.

1, 4, 20, 120, 560, 19600, 27720, 1521520, 7207200, 2845642800, 4170866700, 249466897680, 9117204216120, 1723262134513920, 2525472914524560, 189169152233901840, 1782424363173854400, 28708458878287188000, 15137401000857582807360, 32632841312905676442600, 647550654467707884653760

Offset

1,2

Comments

For known terms > 1:

- a(n) is divisible by a square.

- a(n) is divisible by 20, n > 2.

The record numbers of tetrahedral divisors corresponding to terms a(1)-a(21) are 0, 1, 3, 4, 6, 7, 11, 16, 20, 23, 25, 32, 39, 44, 53, 57, 58, 64, 69, 72, 84.

a(18)..a(21) are divisible by 25878772920. If a(22)..a(28) are divisible by 25878772920 then they are binomial(k + 2, 3) for k in {879207614, 4118478208, 6399801198, 8921309759, 9985690350, 14992913375} having 94, 96, 98, 101, 106, 118 smaller tetrahedral divisors respectively. - David A. Corneth, Mar 22 2021

Links

Table of n, a(n) for n=1..21.

Example

4 is a term because it is the smallest tetrahedral number divisible by the only positive smaller tetrahedral number 1.
20 is a term because it divisible by 1,4,10, and has more divisors than each of 1,4,10, the only smaller terms.

Prog

(PARI) t(n) = n*(n+1)*(n+2)/6;
f(n) = my(tn=n*(n+1)*(n+2)/6); sum(k=1, n-1, (tn % t(k)) == 0);
lista(nn) = {my(nb = - 1, new); for (n=1, nn, new = f(n); if (new > nb, print1(t(n), ", "); nb = new); ); } \\ Michel Marcus, Oct 02 2018

Crossrefs

Cf. A000292, A084260, A318556, A318701.

Subsequence of A013929.

Sequence in context: A101055 A208232 A013197 * A089498 A046729 A277920

Adjacent sequences: A319785 A319786 A319787 * A319789 A319790 A319791

Keyword

nonn

Author

Torlach Rush, Sep 27 2018

Extensions

a(17)-a(18) from Giovanni Resta, Sep 28 2018

a(19)-a(21) from Bert Dobbelaere, Mar 21 2021

Status

approved