cp's OEIS Frontend

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

%I A308852 #23 Jul 30 2019 11:31:41
%S A308852 1,2,3,5,8,16,31,62,129,279,621,1421,3343,8057,19870,50071,128747,
%T A308852 337414,900358,2443947,6742667,18893218,53729800,154983562,453174686,
%U A308852 1342528227,4027584682,12230119228,37574801086,116753643340,366767636286,1164414663338,3734900007009
%N A308852 Minimum number k such that the k-th tetrahedral number is not smaller than n!.
%C A308852 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.
%C A308852 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.
%F A308852 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
%F A308852 a(n) = floor((6*n!)^(1/3)). - _Giovanni Resta_, Jul 30 2019
%e A308852 The minimum tetrahedral number not smaller than 4! is 35 (i.e., the 5th tetrahedral number), so a(4) = 5.
%e A308852 The minimum tetrahedral number not smaller (equal, in fact) than 5! is 120 (i.e., the 8th tetrahedral number), so a(5) = 8.
%t A308852 Floor[(6 Range[33]!)^(1/3)] (* _Giovanni Resta_, Jul 30 2019 *)
%o A308852 (PARI) a(n) = {my(k=1); while (k*(k+1)*(k+2)/6 < n!, k++); k;} \\ _Michel Marcus_, Jun 28 2019
%Y A308852 Cf. A000142 (factorial numbers), A000292 (tetrahedral numbers).
%Y A308852 Cf. A055228 (for n^2), A214049 (for n^3), A214448 (for n^4).
%K A308852 nonn
%O A308852 1,2
%A A308852 _Lorenzo Sauras Altuzarra_, Jun 28 2019
%E A308852 a(26)-a(33) from _Daniel Suteu_, Jun 30 2019