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.

Previous Showing 11-15 of 15 results.

A079714 Number of 2's in n!.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 3, 2, 0, 3, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 1, 0, 0, 4, 7, 3, 1, 4, 3, 3, 3, 7, 4, 5, 4, 3, 4, 4, 4, 8, 6, 6, 10, 3, 10, 3, 6, 9, 6, 1, 9, 10, 6, 9, 10, 13, 8, 6, 11, 8, 8, 8, 14, 7, 8, 10, 8, 14, 9, 12, 10, 16, 8, 12, 9, 5, 9, 12, 14, 17, 16, 12, 9, 10, 8, 8, 17, 11, 19, 7, 13, 16, 19, 19, 14
Offset: 0

Views

Author

Cino Hilliard, Jan 31 2003

Keywords

Links

Crossrefs

Cf. A000142, A137579, A137580.
Cf. A316863.

Programs

  • Maple
    a:= n-> numboccur(2, convert(n!, base, 10)):
seq(a(n), n=0..101);  # Alois P. Heinz, Apr 26 2021
  • Mathematica
    Table[DigitCount[n!,10,2],{n,0,110}] (* Harvey P. Dale, Jun 20 2021 *)
  • PARI
    a(n) = #select(x->(x==2), digits(n!)); \\ Michel Marcus, Apr 26 2021

Formula

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

Extensions

a(36) ff. corrected by Georg Fischer, Apr 26 2021

A137577 Largest of the least frequent digits in decimal representation of n!.

Original entry on oeis.org

9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 8, 9, 6, 9, 6, 1, 9, 9, 5, 8, 9, 9, 5, 7, 7, 9, 9, 2, 7, 9, 4, 7, 8, 8, 6, 8, 8, 6, 6, 9, 7, 9, 9, 8, 7, 7, 9, 9, 9, 9, 9, 1, 6, 8, 2, 7, 7, 5, 5, 1, 4, 5, 7, 1, 5, 7, 6, 6, 3, 9, 5, 9, 6, 3, 1, 5, 9, 4, 9, 8, 5, 8, 5, 8, 8, 6, 7, 4, 3, 3, 9, 9, 9, 1, 2, 5, 1, 7, 7
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 27 2008

Keywords

Comments

A137578(n) <= a(n).

Examples

			n=10: 10! = 3628800 => a(10) = Max{1,4,5,7,9} = 9;
n=11: 11! = 39916800 => a(11) = Max{2,4,5,7} = 7.

Links

Crossrefs

Cf. A000142, A027869, A079680, A079714, A079684, A079688, A079690, A079691, A079692, A079693, A079694, A137579, A137580.

A137578 Smallest of the least frequent digits in decimal representation of n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 7, 5, 3, 3, 1, 5, 6, 7, 2, 2, 2, 7, 3, 4, 2, 1, 8, 5, 8, 8, 5, 3, 9, 2, 4, 9, 8, 3, 7, 9, 9, 5, 4, 1, 1, 5, 7, 2, 3, 6, 5, 5, 1, 4, 5, 7, 1, 5, 3, 6, 6, 3, 4, 5, 9, 6, 3, 1, 5, 9, 4, 9, 3, 5, 8, 4, 8, 1, 6, 7, 4, 2, 3, 9, 9, 9, 1, 2, 4, 1, 7, 7
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 27 2008

Keywords

Comments

a(n) <= A137577(n).

Examples

			n=10: 10! = 3628800 => a(10) = Min{1,4,5,7,9} = 1;
n=11: 11! = 39916800 => a(11) = Min{2,4,5,7} = 2.

Links

Crossrefs

Cf. A000142, A027869, A079680, A079714, A079684, A079688, A079690, A079691, A079692, A079693, A079694, A137579, A137580.

A375348 a(n) is the mode of the digits of n! not counting trailing zeros (using -1 if multimodal).

Original entry on oeis.org

1, 1, 2, 6, -1, -1, -1, -1, -1, 8, 8, 9, 0, 2, -1, -1, 8, -1, 7, -1, -1, -1, 7, 8, 3, 1, 6, 8, -1, -1, 8, 2, 3, 8, 9, -1, 9, -1, 0, 8, 1, -1, -1, 3, 8, 6, -1, 1, 7, 2, 6, -1, 8, 3, -1, 5, 4, 2, -1, 8, 4, 0, 2, 6, -1, 2, 4, 6, 1, 2, 8, 8, 8, 0, 2, 4, -1, 8, 2, 1, 5, 7, 4, -1, 1, 0
Offset: 0

Views

Author

Stefano Spezia and Robert G. Wilson v, Aug 12 2024

Keywords

Comments

Inspired by A356758.
If we were to count trailing zeros, then would have a(n) = 0 for all n >= 34. Therefore we only consider the decimal digits of A004154(n).
Conjecture: excluding -1, as n -> oo, all digits occur equally often.

Examples

			a(0) = a(1) = 1 because 0! = 1! = 1 and 1 is the only digit present;
a(4) = -1 since 4! = 24 and there are only two digits appearing with the same frequency, 2 and 4.
a(14) = -1 because 14! = 87178291200 and, not counting the two trailing 0's, there are two 1's, two 2's, two 7's, and two 8's.

Crossrefs

Cf. A004154, A027869, A031144, A034886, A061010, A137579, A137580, A356758.

Programs

  • Mathematica
    a[n_] := If[Length[c=Commonest[IntegerDigits[n! / 10^IntegerExponent[n!]]]] > 1, -1, c[[1]]]; Array[a, 86, 0]
  • Python
    from collections import Counter
from sympy import factorial
def A375348(n): return -1 if len(k:=Counter(str(factorial(n)).rstrip('0')).most_common(2)) > 1 and k[0][1]==k[1][1] else int(k[0][0]) # Chai Wah Wu, Sep 15 2024

A375575 a(n) is the least frequent digit of n! not counting trailing zeros, or -1 if there is more than one least frequent digit.

Original entry on oeis.org

1, 1, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 2, -1, -1, 7, 0, 4, -1, -1, -1, -1, -1, -1, 8, -1, -1, -1, 8, -1, -1, 9, -1, -1, 0, 9, 9, -1, -1, -1, 1, -1, -1, 2, -1, -1, 5, 5, 1, 4, 5, 7, -1, 5, -1, 6, 6, 0, -1, 5, 9, 6, -1, 0, 5, 9
Offset: 0

Views

Author

Stefano Spezia and Robert G. Wilson v, Aug 19 2024

Keywords

Comments

Analogous to A375348.
If we were to count trailing zeros, then a(n) would never equal zero, for all n's >= 0. Therefore we only consider the decimal digits of A004154.
Conjecture: excluding -1, as n -> oo, the digits distribution is uniform as in A375348.

Examples

			a(0) = a(1) = 1 because 0! = 1! = 1 and 1 is the only digit present;
a(4) = -1 since 4! = 24 and there are two least frequent digits, 2 and 4.
a(14) = 9 because 14! = 87178291200 and, not counting the two trailing 0's, there are two 1's, two 2's, two 7's, two 8's but only one 9.

Crossrefs

Cf. A004154, A027869, A031144, A034886, A061010, A137579, A137580, A375348.

Programs

  • Maple
    f:= proc(n) local L,j;
  L:= convert(n!,base,10);
  for j from 1 while L[j] = 0 do od:
  L:= Statistics:-Tally(L[j...-1]);
  L:= sort(L,(a,b) -> rhs(a) < rhs(b));
  if nops(L) >= 2 and rhs(L[2]) = rhs(L[1]) then -1 else lhs(L[1]) fi
end proc:
map(f, [$0..100]); # Robert Israel, Sep 02 2024
  • Mathematica
    Rarest[lst_] := MinimalBy[ Tally[lst], Last][[All, 1]]; a[n_] := If[ Length[c = Rarest[ IntegerDigits[n!/10^IntegerExponent[n!, 10]] ]] >1, -1, c[[1]]]; Array[a, 80, 0]
  • Python
    from collections import Counter
from sympy import factorial
def A375575(n): return -1 if len(k:=Counter(str(factorial(n)).rstrip('0')).most_common()) > 1 and k[-1][1]==k[-2][1] else int(k[-1][0]) # Chai Wah Wu, Sep 15 2024
Previous Showing 11-15 of 15 results.

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