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

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

0,2

Comments

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 in binary representation n! doesn't contain m!;

A103672(n) = greatest m less than n such that in binary representation n! contains m!.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Index entries for sequences related to factorial numbers.

Index entries for sequences related to binary expansion of n

Example

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'.

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A094700 A073635 A071532 * A165597 A269757 A308973

Adjacent sequences: A102727 A102728 A102729 * A102731 A102732 A102733

Keyword

nonn

Author

Reinhard Zumkeller, Feb 07 2005

Status

approved