A308852 Minimum number k such that the k-th tetrahedral number is not smaller than n!.

1, 2, 3, 5, 8, 16, 31, 62, 129, 279, 621, 1421, 3343, 8057, 19870, 50071, 128747, 337414, 900358, 2443947, 6742667, 18893218, 53729800, 154983562, 453174686, 1342528227, 4027584682, 12230119228, 37574801086, 116753643340, 366767636286, 1164414663338, 3734900007009

Offset

1,2

Comments

More formally, a(n) is the minimum element of the set of positive integers k such that the k-th tetrahedral number is not smaller than the n-th factorial.

Open problem: what is the cardinality of the set of numbers that are simultaneously a tetrahedral number and a factorial number? For example, 1 and 120 belong to this set.

Links

Table of n, a(n) for n=1..33.

Formula

a(n) = ceiling((sqrt(3) * sqrt(243*(n!)^2 - 1) + 27*n!)^(1/3) / 3^(2/3) + 1/(3^(1/3) * (sqrt(3) * sqrt(243*(n!)^2 - 1) + 27*n!)^(1/3)) - 1). - Daniel Suteu, Jun 30 2019

a(n) = floor((6*n!)^(1/3)). - Giovanni Resta, Jul 30 2019

Example

The minimum tetrahedral number not smaller than 4! is 35 (i.e., the 5th tetrahedral number), so a(4) = 5.
The minimum tetrahedral number not smaller (equal, in fact) than 5! is 120 (i.e., the 8th tetrahedral number), so a(5) = 8.

Mathematica

Floor[(6 Range[33]!)^(1/3)] (* Giovanni Resta, Jul 30 2019 *)

Prog

(PARI) a(n) = {my(k=1); while (k*(k+1)*(k+2)/6 < n!, k++); k; } \\ Michel Marcus, Jun 28 2019

Crossrefs

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

Cf. A055228 (for n^2), A214049 (for n^3), A214448 (for n^4).

Sequence in context: A306622 A030034 A370002 * A093000 A122630 A108054

Adjacent sequences: A308849 A308850 A308851 * A308853 A308854 A308855

Keyword

nonn

Author

Lorenzo Sauras Altuzarra, Jun 28 2019

Extensions

a(26)-a(33) from Daniel Suteu, Jun 30 2019

Status

approved