cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A035066 Prime lengths of factorials: see A035065.

Original entry on oeis.org

2, 3, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 47, 53, 67, 79, 97, 101, 127, 131, 137, 139, 149, 167, 173, 179, 181, 191, 193, 197, 199, 227, 229, 233, 239, 257, 263, 281, 283, 307, 337, 359, 373, 389, 401, 431, 443, 457, 467, 479, 491, 503, 541, 563, 571, 593
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A000142, A035065.

Programs

  • Mathematica
    Select[Table[IntegerLength[n!],{n,500}],PrimeQ] (* Harvey P. Dale, May 03 2015 *)

Extensions

Offset changed to 1 by Giovanni Resta, Oct 08 2019

A034886 Number of digits in n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 102
Offset: 0

Views

Author

Erich Friedman

Keywords

Comments

Most counterexamples to the Kamenetsky formula (see below) must belong to A177901.
Noam D. Elkies reported on MathOverflow (see link):
"A counterexample [to Kamenetsky's formula] is n_1 := 6561101970383, with log_10((n_1/e)^n_1*sqrt(2*Pi*n_1)) = 81244041273652.999999999999995102482, but log_10(n_1!) = 81244041273653.000000000000000618508. [...] n_1 is the first counterexample, and the only one up to 10^14."
From Bernard Schott, Dec 07 2019: (Start)
a(n) < n iff 2 <= n <= 21;
a(n) = n iff n = 1, 22, 23, 24;
a(n) > n iff n = 0 or n >= 25. (End)

References

  • Martin Gardner, "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978

Links

Crossrefs

Cf. A000142, A055642.
Cf. A006488 (a(n) is a square), A056851 (a(n) is a cube), A035065 (a(n) is a prime), A333431 (a(n) is a factorial), A333598 (a(n) is a palindrome), A067367 (p and a(p) are primes), A058814 (n divides a(n)).
Cf. A137580 (number of distinct digits in n!), A027868 (number of trailing zeros in n!).

Programs

  • Haskell
    a034886 = a055642 . a000142  -- Reinhard Zumkeller, Apr 08 2012
  • Magma
    [Floor(Log(Factorial(n))/Log(10)) + 1: n in [0..30]]; // G. C. Greubel, Feb 26 2018
  • Maple
    A034886 := n -> `if`(n<2,1,`if`(n<6561101970383, ceil((ln(2*Pi)-2*n+ln(n)*(1+2*n))/(2*ln(10))),length(n!))); # Peter Luschny, Aug 26 2011
  • Mathematica
    Join[{1, 1}, Table[Ceiling[Log[10, 2 Pi n]/2 + n*Log[10, n/E]], {n, 2, 71}]]
f[n_] := Floor[(Log[2Pi] - 2n + Log[n]*(1 + 2n))/(2Log[10])] + 1; f[0] = f[1] = 1; Array[f, 72, 0] (* Robert G. Wilson v, Jan 09 2013 *)
IntegerLength/@(Range[0,80]!) (* Harvey P. Dale, Aug 07 2022 *)
  • PARI
    for(n=0,30, print1(floor(log(n!)/log(10)) + 1, ", ")) \\ G. C. Greubel, Feb 26 2018

Formula

a(n) = floor(log(n!)/log(10)) + 1.
a(n) = A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n); a(n) = A055642(A000142(n)). - Reinhard Zumkeller, Jan 27 2008
Using Stirling's formula we can derive an approximation, which is very fast to compute in practice: ceiling(log_10(2*Pi*n)/2 + n*(log_10(n/e))). This approximation gives the exact answer for 2 <= n <= 5*10^7. - Dmitry Kamenetsky, Jul 07 2008
a(n) = ceiling(log_10(1) + log_10(2) + ... + log_10(n)). - Dmitry Kamenetsky, Nov 05 2010

Extensions

Explained that the formula is an approximation. Made the formula easier to read. - Dmitry Kamenetsky, Dec 15 2010

A067367 Primes p such that the number of digits in p! is also prime.

Original entry on oeis.org

5, 23, 29, 43, 89, 107, 139, 193, 199, 211, 229, 239, 313, 347, 419, 457, 587, 641, 661, 673, 701, 739, 863, 877, 1091, 1097, 1103, 1217, 1259, 1283, 1361, 1381, 1447, 1493, 1549, 1607, 1609, 1663, 1693, 1741, 1861, 1973, 1987, 2083, 2251, 2351, 2399, 2423
Offset: 1

Views

Author

Sudipta Das (juitech(AT)vsnl.net), Feb 23 2002

Keywords

Examples

			a(1)=5 because 5!=120 has 3 (prime) digits.

Links

Crossrefs

Cf. A035065.

Programs

  • Mathematica
    Select[ Select[ Range[2500], PrimeQ[ Floor[ Log[10, #! ] + 1]] & ], PrimeQ[ # ] & ]
Select[Prime[Range[400]],PrimeQ[IntegerLength[#!]]&] (* Harvey P. Dale, Mar 08 2018 *)

Extensions

Edited and extended by Robert G. Wilson v, Feb 28 2002

A333598 Numbers m such that m! has a palindromic number of decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 22, 30, 37, 44, 57, 63, 69, 70, 81, 86, 91, 106, 111, 116, 126, 131, 140, 145, 154, 163, 168, 177, 186, 199, 221, 225, 238, 242, 255, 259, 288, 292, 368, 372, 384, 388, 407, 411, 419, 423, 438, 450, 532
Offset: 1

Views

Author

Bernard Schott, Mar 28 2020

Keywords

Comments

The corresponding palindromic numbers are 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 77, 88, 99, 101, ...
Nice result: 22 is a palindrome and 22! has 22 digits, and also, 44! has 55 digits.

Examples

			14! = 87178291200 that has 11 digits, 11 is a palindrome, hence 14 is a term.

Crossrefs

Cf. A000142, A002113, A034886.
Cf. A006488 (similar, with square), A035065 (similar, with prime), A056851 (similar, with cube), A333431 (similar, with factorial).

Programs

  • Mathematica
    Select[Range[0, 532], PalindromeQ @ Length @ IntegerDigits[#!] &] (* Amiram Eldar, Mar 28 2020 *)
Select[Range[0,550],PalindromeQ[IntegerLength[#!]]&] (* Harvey P. Dale, Oct 30 2023 *)
  • PARI
    isok(m) = my(d=digits(#Str(m!))); d == Vecrev(d); \\ Michel Marcus, Mar 28 2020
Showing 1-4 of 4 results.

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