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 1-5 of 5 results.

A094016 Complement of A092601.

Original entry on oeis.org

5, 10, 11, 13, 22, 25, 29, 31, 32, 36, 41, 47, 49, 50, 52, 54, 57, 60, 67, 69, 72, 76, 79, 88, 90, 92, 98, 99, 102, 104, 105, 107, 109, 110, 112, 113, 121, 122, 125, 130, 135, 141, 143, 144, 148, 150, 151, 154, 160, 163, 165, 168, 170, 171, 174, 179, 182, 184, 186
Offset: 1

Views

Author

Robert G. Wilson v, Apr 21 2004

Keywords

Crossrefs

Cf. A092601.

Programs

  • Mathematica
    f[n_] := ToString[ FromDigits[ IntegerDigits[n, 2]]]; g[n_] := g[n] = Block[{c = 0, k = 1, s = f[n! ]}, While[k <= n, If[ StringPosition[ s, f[k]] != {}, c++ ]; k++ ]; c]; Complement[ Range[200], Union[ Table[ g[n], {n, 202}]]]

A093684 In binary representation: number of occurrences of n in n!.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 1, 3, 0, 2, 1, 3, 1, 1, 0, 1, 2, 2, 3, 3, 1, 1, 1, 1, 2, 2, 4, 3, 3, 3, 2, 2, 0, 3, 1, 5, 5, 6, 4, 1, 5, 2, 3, 2, 2, 4, 1, 1, 1, 4, 1, 1, 1, 2, 3, 3, 4, 5, 0, 3, 2, 1, 4, 3, 4, 5, 3, 2, 1, 2, 3, 3, 3, 3, 6, 2, 3, 4, 4, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 10 2004

Keywords

Comments

a(A093685(n)) = 0, a(A093686(n)) > 0.

Examples

			n=12->'1100', 12!=479001600->'11100100011001111110000000000' with three occurrences of '1100': '.1100....1100....1100........', therefore a(12)=3.

Links

Crossrefs

Cf. A036603, A007088, A000142, A092601.

Programs

  • Maple
    f:= proc(n) local L,Lf;
  L:= convert(convert(n,binary),string);
  Lf:= convert(convert(n!,binary),string);
  nops([StringTools:-SearchAll(L,Lf)])
end proc:
map(f, [$1..100]); # Robert Israel, May 20 2016
  • Mathematica
    non[n_]:=Module[{b=IntegerDigits[n,2],f=IntegerDigits[n!,2]}, Length[ Select[ Partition[ f,Length[b],1],#==b&]]]; Array[non,110] (* Harvey P. Dale, Jun 04 2014 *)

A093710 Numbers k such that in their binary representation all numbers from 1 to k are contained in k!.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 14, 23, 26, 28, 30, 33, 34, 35, 39, 42, 43, 51, 53, 58, 61, 62, 63, 64, 66, 68, 70, 73, 77, 80, 83, 93, 94, 106, 108, 111, 114, 115, 116, 126, 131, 132, 133, 134, 136, 137, 147, 149, 153, 155, 156, 169, 172, 175, 180, 185, 187, 191, 195, 206
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2004

Keywords

Examples

			6 is in the sequence because 6! = 1011010000_2 which contains 4 = 100_2, 5 = 101_2 and 6 = 110_2 as a substring in the binary expansion. As it contains 4, 5 and 6 in binary it contains the binary expansion of every smaller number than 4 in its binary expansion. - _David A. Corneth_, Aug 05 2025

Links

Crossrefs

Complement of A093711.
Cf. A000142, A007088, A036603, A072831, A092601.

Formula

A092601(a(n)) = a(n).

A093711 Numbers k such that in their binary representation not all numbers from 1 to k are contained in k!.

Original entry on oeis.org

5, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 32, 36, 37, 38, 40, 41, 44, 45, 46, 47, 48, 49, 50, 52, 54, 55, 56, 57, 59, 60, 65, 67, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2004

Keywords

Links

Crossrefs

Complement of A093710.
A093685 \ {0} is a subsequence.
Cf. A000142, A007088, A036603, A092601.

Formula

A092601(a(n)) < a(n).

A092762 Least number whose binary representation is not contained in that of n!.

Original entry on oeis.org

2, 3, 4, 5, 5, 7, 10, 10, 7, 9, 7, 5, 13, 15, 15, 15, 17, 17, 15, 17, 18, 17, 29, 21, 17, 28, 20, 31, 21, 36, 15, 15, 35, 42, 41, 35, 33, 22, 45, 22, 36, 46, 50, 41, 37, 33, 21, 40, 38, 31, 66, 33, 60, 45, 51, 42, 42, 60, 35, 38, 63, 65, 75, 75, 39, 71, 36, 75, 67, 74, 67, 68, 78
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 13 2004

Keywords

Comments

a(A093710(n)) > n, a(A093711(n)) <= n.
Sum_{n=1..3000} a(n) = 4179226. The average value of a(n)/n -> 0.93+ and except for some initial terms, < 212, 0.90 < a(n)/n < 0.95. But there is graphical evidence that this value may increase. - Robert G. Wilson v, Apr 21 2004
a(n) = n only for 5, 13, 15, 17, 513, 517, 1041, 1538, 2138, n <= 2500. a(n) = n+1 for n=1, 2, 3, 4, 6, 14, 134, 137, 155, 169, 216, 313, 530, 1035 & 2402. - Robert G. Wilson v, Apr 21 2004
a(n) >= 3n/2 for n=1, 2, 334, 335, 2693 & 2739. - Robert G. Wilson v, Apr 21 2004
a(n) <= n/2 for n=12, 31, 32, 47, 122, 142, 152, 188, 303, 378, 443, 548, 598, 1319, 1354, 1420, 2127, 2137, 2223, 2230, 2368, 2433, 2571. - Robert G. Wilson v, Apr 21 2004

Examples

			a(7)=10 because 7! = 1001110110000_b and 1_b, 10_b, 11_b, 100_b, 101_b, 110_b, 111_b, 1000_b & 1001_b are substrings, but 10_d = 1010_b is not a substring. - _Robert G. Wilson v_, Apr 21 2004

Crossrefs

Cf. A036603, A007088, A000142, A092601.

Programs

  • Mathematica
    f[n_] := ToString[ FromDigits[ IntegerDigits[n, 2]]]; g[n_] := g[n] = Block[{k = 1, s = f[n! ]}, While[ StringPosition[ s, f[k]] != {}, k++ ]; k]; Table[ g[n], {n, 75}] (* Robert G. Wilson v, Apr 21 2004 *)

Extensions

Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.