A214046 -id:A214046 - cp's OEIS frontend

A340810 Triangle T(n,k), n>=2, 2 <= k <= A214046(n), read by rows, where T(n,k) = n! mod k^n.

2, 6, 8, 24, 24, 120, 16, 720, 48, 666, 5040, 128, 954, 40320, 384, 8586, 100736, 362880, 768, 26811, 483072, 3628800, 1280, 58725, 2168064, 39916800, 3072, 173259, 9239552, 234860975, 479001600

Offset: 2

Seiichi Manyama, Jan 22 2021

			n\k  |    2       3        4          5          6
-----+---------------------------------------------
   2 |    2;
   3 |    6;
   4 |    8,     24;
   5 |   24,    120;
   6 |   16,    720;
   7 |   48,    666,    5040;
   8 |  128,    954,   40320;
   9 |  384,   8586,  100736,    362880;
  10 |  768,  26811,  483072,   3628800;
  11 | 1280,  58725, 2168064,  39916800;
  12 | 3072, 173259, 9239552, 234860975, 479001600;

Seiichi Manyama, Rows n = 2..200, flattened

Column k=2..4 give A068496, A212309, A212310.

Cf. A000142, A074184, A214046.

row[n_] := Module[{k = 1, s = {}}, While[k^n <= n!, k++; AppendTo[s, Mod[n!, k^n]]]; s]; Table[row[n], {n, 2, 12}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def A(n)
  m = f(n)
  ary = []
  (2..n).each{|i|
    j = i ** n
    ary << m % j
    break if m <= j
  }
  ary
end
def A340810(n)
  (2..n).map{|i| A(i)}.flatten
end
p A340810(12)

A087375 Smallest n-th power > n!.

Original entry on oeis.org

2, 4, 8, 81, 243, 729, 16384, 65536, 1953125, 9765625, 48828125, 2176782336, 13060694016, 678223072849, 4747561509943, 33232930569601, 2251799813685248, 18014398509481984, 144115188075855872, 12157665459056928801, 109418989131512359209, 10000000000000000000000, 100000000000000000000000, 1000000000000000000000000, 108347059433883722041830251

Offset: 1

Amarnath Murthy, Sep 09 2003

nonn

			a(5) = 243 as 243 = 3^5 > 5! > 2^5.

Robert Israel, Table of n, a(n) for n = 1..448

Cf. A087374, A214046.

g:= proc(n)  floor(1+(n!)^(1/n))^n end proc:
map(g, [$1..30]); # Robert Israel, Nov 25 2024

a[n_]:=Floor[1+(n!)^(1/n)]^n; Array[a,25] (* Stefano Spezia, Mar 27 2025 *)

a(n) = A214046(n)^n for n > 1. - Robert Israel, Nov 25 2024

Corrected and extended by Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004

A214449 Least m>0 such that n! <= m^m.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 52

Offset: 1

Clark Kimberling, Jul 18 2012

Conjecture: this sequence results from A118168 by deleting the first two 1s.

			a(4) = 3 because 2^2 < 4! < 3^3.

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

Cf. A214046.

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

A306904 The geometric mean of the first n integers, rounded to the nearest integer.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25

Offset: 1

Robin Powell, Mar 15 2019

nonn

a(n) is the least k such that (k-1/2)^n < n!. - Robert Israel, May 05 2019

			a(5) is the 5th root of the product of the first 5 integers (approx. 2.605171) rounded up to 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000142, A061375 (floor instead of round), A214046.

Res:= 1: last:= 1: v:= 1:
for n from 2 to 100 do
  v:= n*v; t:= 2^n*v;
  for k from last while (2*k-1)^n < t do od:
  last:= k-1;
  Res:= Res, last;
od:
Res; # Robert Israel, May 05 2019

Array[Round[#!^(1/#)] &, 68] (* Michael De Vlieger, Mar 31 2019 *)
Table[Round[GeometricMean[Range[n]]],{n,70}] (* Harvey P. Dale, Mar 19 2023 *)

a(n) = round(n!^(1/n)); \\ Michel Marcus, May 05 2019

from sympy import integer_nthroot, factorial
def A306904(n): return (m:=int(integer_nthroot(f:=int(factorial(n)),n)[0]))+int((f<=((m<<1)+1)**n) # Chai Wah Wu, Jun 06 2025

a(n) = round(n!^(1/n)).

a(n) ~ n/e + log(n)/(2*e). - Robert Israel, May 05 2019

Corrected by Robert Israel, May 05 2019

cp's OEIS Frontend

A340810 Triangle T(n,k), n>=2, 2 <= k <= A214046(n), read by rows, where T(n,k) = n! mod k^n.

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A087375 Smallest n-th power > n!.

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A214449 Least m>0 such that n! <= m^m.

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Mathematica

A306904 The geometric mean of the first n integers, rounded to the nearest integer.

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