

A308852


Minimum number k such that the kth tetrahedral number is not smaller than n!.


1



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
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OFFSET

1,2


COMMENTS

More formally, a(n) is the minimum element of the set of positive integers k such that the kth tetrahedral number is not smaller than the nth 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



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


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



PROG

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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



