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A102730
Number of factorials contained in n! in binary representation.
13
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!.
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 07 2005
STATUS
approved