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 A308852 Minimum number k such that the k-th 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 (list; graph; refs; listen; history; text; internal format)
 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: A121649 A306622 A030034 * 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

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Last modified November 29 03:03 EST 2023. Contains 367422 sequences. (Running on oeis4.)