A213857 -id:A213857 - cp's OEIS frontend

A127110 n! in base 3.

1, 1, 2, 20, 220, 11110, 222200, 20220200, 2001022100, 200102210000, 20211100210000, 2210002222120000, 1020101022210200000, 121001222020102200000, 22100000121100001100000, 11122000100002000202000000, 2202002012101012102212000000, 1201122101101102102110111000000

Offset: 0

Artur Jasinski, Jan 05 2007

nonn

Cf. A000142, A007089.

Cf. A136754, A136690, A125552, A213857.

Table[FromDigits[IntegerDigits[n!,3]],{n,0,17}] (* Stefano Spezia, Aug 25 2022 *)

a(n) = fromdigits(digits(n!, 3), 10); \\ Michel Marcus, Sep 04 2021

A213858 Least m such that n! <= 4^m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 38, 40, 42, 45, 47, 49, 52, 54, 57, 59, 62, 64, 67, 70, 72, 75, 77, 80, 83, 85, 88, 91, 94, 96, 99, 102, 105, 108, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 145, 148

Offset: 1

Clark Kimberling, Jul 17 2012

			a(11) = 13 because 4^12 < 11! <= 4^13.

Harvey P. Dale, Table of n, a(n) for n = 1..1000 (Identical to file previously submitted by Clark Kimberling except for first term.)

Cf. A213857.

Ceiling[Log[4,Range[100]!]] (* Harvey P. Dale, Jul 21 2016 *)

Corrected (a(1) changed to zero) by Harvey P. Dale, Jul 21 2016

A214047 Least m>0 such that n! <= (3/2)^m.

Original entry on oeis.org

1, 2, 5, 8, 12, 17, 22, 27, 32, 38, 44, 50, 56, 63, 69, 76, 83, 90, 98, 105, 112, 120, 128, 136, 144, 152, 160, 168, 176, 185, 193, 202, 210, 219, 228, 237, 245, 254, 263, 273, 282, 291, 300, 310, 319, 328, 338, 347, 357, 367

Offset: 1

Clark Kimberling, Jul 18 2012

			a(6) = 17 because (3/2)^16 < 6! <= (3/2)^17.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A003070, A213857.

Table[m=1; While[n!>(3/2)^m, m++]; m, {n,1,100}]
Join[{1},With[{c=Log[3/2]},Table[Ceiling[Log[n!]/c],{n,2,50}]]] (* Harvey P. Dale, May 15 2013 *)

A214048 Least m>0 such that n! <= r^m, where r = (1+sqrt(5))/2, the golden ratio.

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 18, 23, 27, 32, 37, 42, 47, 53, 58, 64, 70, 76, 82, 88, 95, 101, 108, 114, 121, 128, 135, 142, 149, 156, 163, 170, 177, 185, 192, 199, 207, 214, 222, 230, 237, 245, 253, 261, 269, 277, 285, 293, 301, 309

Offset: 1

Clark Kimberling, Jul 18 2012

Also, the least m>0 such that n! < L(m), where L = A000032, the Lucas numbers.

			a(4) = 7 because r^6 < 4! <= 4^7.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Aleksandar Petojević, Lambert's W function and Kurepa's left factorial, Project: Kurepa's hypothesis for left factorial, ResearchGate (2023).
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7.

Cf. A000032, A001622, A003070, A213857, A214047.

Table[m=1; While[n!>GoldenRatio^m, m++]; m, {n,1,100}]

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A127110 n! in base 3.

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A213858 Least m such that n! <= 4^m.

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A214047 Least m>0 such that n! <= (3/2)^m.

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A214048 Least m>0 such that n! <= r^m, where r = (1+sqrt(5))/2, the golden ratio.

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