A230319 -id:A230319 - cp's OEIS frontend

A336803 Numbers k such that A230319(k) - A230319(k-1) = 2.

3, 6, 9, 12, 15, 19, 22, 26, 30, 34, 38, 43, 47, 51, 56, 60, 65, 70, 75, 79, 84, 89, 94, 99, 104, 110, 115, 120, 125, 130, 136, 141, 147, 152, 158, 163, 169, 174, 180, 185, 191, 197, 202, 208, 214, 220, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291

Offset: 1

Enrique Treviño, Jan 27 2021

nonn

Sánchez Garza and Treviño proved the counting function for this sequence is asymptotic to x/log x. See link.

			3 is a term because A230319(3) - A230319(2) = 2.

M. Sánchez Garza and E. Treviño, On a sequence related to the factoradic representation of an integer, Journal of Integer Sequences Vol. 24 (2021), Article 21.8.5.

Cf. A230319.

j[r_] := j[r] = Module[{k = 1}, While[k! <= k^(r - 1), k++]; k];
jPrimes = {}; Do[If[j[r + 1] - j[r] == 2, AppendTo[jPrimes, r], 0], {r, 1, 2500}]

f(n) = my(k=1); while (k^n >= k!, k++); k; \\ A230319
isok(n) = f(n) - f(n-1) == 2; \\ Michel Marcus, Jan 27 2021

A230282 Largest k such that (k*n)! >= (k!)^(n+1).

Original entry on oeis.org

1, 1, 6, 64, 679, 8468, 126784, 2238565, 45605124, 1053117974, 27182818156, 775557529509, 24236473829015, 823299898542083, 30205566231626957, 1190319005015526817, 50143449209799256306, 2248672171655330927835

Offset: 0

Alex Ratushnyak, Oct 14 2013

nonn

			Biggest k such that (3*k)! >= k!^4 is k = 64, so a(3) = 64.
a(10) = 27182818156 because k = 27182818156 satisfies the inequality (k*10)! >= (k!)^11, but k = 27182818157 does not. To verify this, note that taking the logarithm of each side of the inequality gives log((k*10)!) >= 11*log(k!), and use the series expression log(m!) = log(2*Pi*m)/2 + m*log(m) - m + (1/12)/m - (1/360)/m^3 + (1/1260)/m^5 - ... (where the numerators and denominators of the fractions 1/12, -1/360, 1/1260, etc., are from A046968 and A046969, respectively), to get, at k = 27182818156, log(271828181560!) = 6884982704601.26... for the left hand side of the inequality, and the slightly smaller result 11*log(27182818156!) = 6884982704600.83... for the right hand side; then repeat the calculations using k = 27182818157, and observe that this makes the right hand side slightly larger than the left hand side. - _Jon E. Schoenfield_, Oct 23 2013

Cf. A000142, A065027, A136432, A230319.

Table[k = 0; While[(k n)! >= (k!)^(n + 1), k++]; k - 1, {n, 0, 4}] (* T. D. Noe, Oct 18 2013 *)

import math
for n in range(8):
  for k in range(10000000):
    if math.factorial(n*k) < math.factorial(k)**(n+1):
      print(k-1, end=', ')
      break

For n > 1, a(n) = floor(e*(n^n) - ((n^2-1)*log(n) + n*(1+log(2*Pi)))/2) [conjectural, but verified for all n in 2..5000]. - Jon E. Schoenfield, Oct 22 2013

a(7)-a(17) from Jon E. Schoenfield, Oct 22 2013

cp's OEIS Frontend

A336803 Numbers k such that A230319(k) - A230319(k-1) = 2.

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