A036603 -id:A036603 - cp's OEIS frontend

A103676 Numbers m such that in binary representation m! contains 5!.

5, 10, 12, 22, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Reinhard Zumkeller, Feb 12 2005

Complement of A103677: A103673(a(n))=1, A103673(A103677(n))=0.

Cf. A102730, A036603, A007088, A000142, A103678, A103680.

Select[Range[100],MemberQ[Partition[IntegerDigits[#!,2],7,1],{1,1,1,1,0,0,0}]&] (* Harvey P. Dale, Apr 09 2012 *)

is(n)=n=n!; while(n>119, my(e=valuation(n, 2), e1=valuation((n>>=e)+1, 2)); n>>=e1; if(e>2 && e1>3, return(1))); 2 \\ Charles R Greathouse IV, Apr 07 2013

A103677 Numbers m such that in binary representation m! doesn't contain 5!.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 30, 31, 32, 36, 46, 53, 58, 65

Offset: 1

Reinhard Zumkeller, Feb 12 2005

Conjecture: there are no further terms;

complement of A103676: A103673(a(n))=0, A103673(A103676(n))=1.

Cf. A102730, A036603, A007088, A000142, A103679, A103681.

is(n)=n=n!; while(n>119, my(e=valuation(n, 2), e1=valuation((n>>=e)+1, 2)); n>>=e1; if(e>2 && e1>3, return(0))); 1 \\ Charles R Greathouse IV, Apr 07 2013

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

Reinhard Zumkeller, Apr 10 2004

nonn

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

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

Cf. A036603, A007088, A000142, A092601.

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

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

Reinhard Zumkeller, Apr 11 2004

			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

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3773 terms from Amiram Eldar)
David A. Corneth, Plot of a(n+1)-a(n), n = 1..9999
David A. Corneth, plot of a(n)/n, n = 1..10000

Complement of A093711.

Cf. A000142, A007088, A036603, A072831, A092601.

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

Reinhard Zumkeller, Apr 11 2004

Amiram Eldar, Table of n, a(n) for n = 1..6227 (terms not exceeding 10^4)

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

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

A127113 n! in base 6.

Original entry on oeis.org

1, 1, 2, 10, 40, 320, 3200, 35200, 510400, 11440000, 205440000, 3543320000, 115310400000, 2505522400000, 104014341200000, 2440423132000000, 112255444052000000, 3300252314304000000, 143012413513200000000, 5313433311353200000000, 302523154413030400000000

Offset: 0

Artur Jasinski, Jan 05 2007

Cf. A127109, A127110, A036603, A127112, A127113, A127114, A127115, A127116.

FromDigits[IntegerDigits[#,6]]&/@(Range[0,30]!) (* Harvey P. Dale, Jun 18 2019 *)

a(n) = A007092(A000142(n)). - R. J. Mathar, Jun 09 2020

More terms from Harvey P. Dale, Jun 18 2019

A127114 n! in base 7.

Original entry on oeis.org

1, 1, 2, 6, 33, 231, 2046, 20460, 225360, 3040650, 42562410, 663200340, 14604306060, 310211542410, 6204234151200, 163322250505200, 4256423144450400, 134630366022322500, 3634363143602406600, 134462435323300144200, 4233013654405404511500

Offset: 0

Artur Jasinski, Jan 05 2007

Number of digits in A127114(n) = A127033(n) + 1.

Cf. A127109, A127110, A036603, A127112, A127113, A127114, A127115, A127116, A127033.

b = Table[IntegerDigits[n!, 7], {n, 1, 15}]; a = {}; Do[AppendTo[a, FromDigits[b[[x]]]], {x, 1, Length[b]}]; a (*Artur Jasinski*)
FromDigits[IntegerDigits[#,7]]&/@(Range[0,20]!) (* Harvey P. Dale, Jun 19 2011 *)

More terms from Harvey P. Dale, Jun 19 2011

A127115 n! in base 8.

Original entry on oeis.org

1, 1, 2, 6, 30, 170, 1320, 11660, 116600, 1304600, 15657400, 230212400, 3443176000, 56312146000, 1211416624000, 23016735654000, 460356735300000, 12067735663300000, 265756631234600000, 6601271140642200000

Offset: 0

Artur Jasinski, Jan 05 2007

nonn

Cf. A127109, A127110, A036603, A127112, A127113, A127114, A127115, A127116.

b = Table[IntegerDigits[n!, 8], {n, 1, 15}]; a = {}; Do[AppendTo[a, FromDigits[b[[x]]]], {x, 1, Length[b]}]; a (*Artur Jasinski*)

A127116 n! in base 9.

Original entry on oeis.org

1, 1, 2, 6, 26, 143, 880, 6820, 61270, 612700, 6740700, 83088500, 1211283600, 17058212600, 270017301300, 4560302022000, 82065335385000, 1648341372414000, 34076827548280000, 725735500635080000, 17178403115182870000

Offset: 0

Artur Jasinski, Jan 05 2007

nonn

Cf. A127109, A127110, A036603, A127112, A127113, A127114, A127115, A127116.

b = Table[IntegerDigits[n!, 9], {n, 1, 15}]; a = {}; Do[AppendTo[a, FromDigits[b[[x]]]], {x, 1, Length[b]}]; a (*Artur Jasinski*)

A093686 In binary representation: numbers occurring at least once in their factorial.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Reinhard Zumkeller, Apr 10 2004

nonn

A093684(a(n)) > 0, complement of A093685.

Almost all numbers are included -- of the first 1000 numbers, only 14 -- i.e., 5, 11, 13, 15, 17, 31, 37, 55, 81, 164, 395, 513, 517, and 619 -- do not appear. In all likelihood, the density of such exceptions gets even smaller as the numbers get larger. - Harvey P. Dale, May 16 2025

			6!=1*2*3*4*5*6=720 -> '1011010000' where '110'=6 is contained:
'..110.....', therefore 6 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A036603, A007088, A000142.

Select[Range[80],SequenceCount[IntegerDigits[#!,2],IntegerDigits[#,2]]>0&] (* Harvey P. Dale, May 16 2025 *)

cp's OEIS Frontend

A103676 Numbers m such that in binary representation m! contains 5!.

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A103677 Numbers m such that in binary representation m! doesn't contain 5!.

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

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

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

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A127113 n! in base 6.

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A127114 n! in base 7.

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A127115 n! in base 8.

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A127116 n! in base 9.

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A093686 In binary representation: numbers occurring at least once in their factorial.

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