A093684 -id:A093684 - cp's OEIS frontend

A102730 Number of factorials contained in the binary representation of n!

1, 2, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 6, 6, 7, 6, 6, 6, 7, 6, 7, 8, 6, 7, 6, 7, 6, 7, 7, 7, 8, 7, 7, 7, 6, 8, 7, 7, 7, 7, 7, 8, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 8, 7, 7, 8, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 8, 7, 7, 7, 7, 8, 7, 7, 8, 8, 7, 7, 7, 8, 8, 7, 8, 7, 7

Offset: 0

Reinhard Zumkeller, Feb 07 2005

Conjecture: the sequence is bounded.
I conjecture the contrary: for every k, there exists n with a(n) > k. See A103670. - Charles R Greathouse IV, Aug 21 2011
For n > 0: A103670(n) = smallest m such that a(m) = n.
A103671(n) = smallest m such that the binary representation of n! does not contain m!.
A103672(n) = greatest m less than n such that the binary representation of n! contains m!.

			n = 6: 6! = 720 -> '1011010000' contains a(6) = 5 factorials: 0! = 1 -> '1', 1! = 1 -> '1', 2! = 2 -> '10', 3! = 6 -> '110' and 6! itself, but not 4! = 24-> '11000' and 5! = 120 -> '1111000'.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n.
Index entries for sequences related to factorial numbers.

Cf. A036603, A007088, A000142, A011371, A093684, A103673, A103676, A103677, A103674, A103678, A103679, A103675, A103680, A103681.

a[n_] := Sum[Boole[StringContainsQ[IntegerString[n!, 2], IntegerString[k!, 2]]], {k, 0, n}]; Array[a, 100, 0] (* Amiram Eldar, Apr 03 2025 *)

contains(v,u)=for(i=0,#v-#u,for(j=1,#u,if(v[i+j]!=u[j],next(2)));return(1));0
a(n)=my(v=binary(n--!));sum(i=0,n-1,contains(v,binary(i!)))+1 \\ Charles R Greathouse IV, Aug 21 2011

A093685 In binary representation: numbers not occurring in their factorial.

Original entry on oeis.org

0, 5, 11, 13, 15, 17, 31, 37, 55, 81, 164, 395, 513, 517, 619, 1041, 1287, 1538, 2108, 2116, 2137, 2138, 2282, 2352, 2363, 2432, 2466, 2524, 4278, 4511, 4758, 4766, 4852, 4854, 5136, 5586, 8396, 8463, 8883, 9707, 10351, 16528, 17279, 19469, 21244, 24472

Offset: 1

Reinhard Zumkeller, Apr 10 2004

A093684(a(n)) = 0, complement of A093686.

			5! = 1*2*3*4*5 = 120 -> '1111000', in which '101'=5 is not contained, so 5 is in the sequence.

Cf. A036603, A007088, A000142, A093826.

f[n_] := ToString[ FromDigits[ IntegerDigits[n, 2]]]; Select[ Range[ 29000], StringPosition[ f[ #! ], f[ # ]] == {} &] (* Robert G. Wilson v, Apr 15 2004 *)

def aupto(limit):
  kfact, alst = 1, [0]
  for k in range(1, limit+1):
    kb, kfact = bin(k)[2:], kfact * k
    kfactb = bin(kfact)[2:]
    if kb not in kfactb: alst.append(k)
  return alst
print(aupto(25000)) # Michael S. Branicky, May 01 2021

More terms from Robert G. Wilson v, Apr 15 2004

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

A102730 Number of factorials contained in the binary representation of n!

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A093685 In binary representation: numbers not occurring in their factorial.

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

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Mathematica