cp's OEIS Frontend

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

%I A102730 #27 Apr 03 2025 02:51:31
%S A102730 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,
%T A102730 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,
%U A102730 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
%N A102730 Number of factorials contained in the binary representation of n!
%C A102730 Conjecture: the sequence is bounded.
%C A102730 I conjecture the contrary: for every k, there exists n with a(n) > k. See A103670. - _Charles R Greathouse IV_, Aug 21 2011
%C A102730 For n > 0: A103670(n) = smallest m such that a(m) = n.
%C A102730 A103671(n) = smallest m such that the binary representation of n! does not contain m!.
%C A102730 A103672(n) = greatest m less than n such that the binary representation of n! contains m!.
%H A102730 Charles R Greathouse IV, <a href="/A102730/b102730.txt">Table of n, a(n) for n = 0..1000</a>
%H A102730 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A102730 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%e A102730 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'.
%t A102730 a[n_] := Sum[Boole[StringContainsQ[IntegerString[n!, 2], IntegerString[k!, 2]]], {k, 0, n}]; Array[a, 100, 0] (* _Amiram Eldar_, Apr 03 2025 *)
%o A102730 (PARI) contains(v,u)=for(i=0,#v-#u,for(j=1,#u,if(v[i+j]!=u[j],next(2)));return(1));0
%o A102730 a(n)=my(v=binary(n--!));sum(i=0,n-1,contains(v,binary(i!)))+1 \\ _Charles R Greathouse IV_, Aug 21 2011
%Y A102730 Cf. A036603, A007088, A000142, A011371, A093684, A103673, A103676, A103677, A103674, A103678, A103679, A103675, A103680, A103681.
%K A102730 nonn,base
%O A102730 0,2
%A A102730 _Reinhard Zumkeller_, Feb 07 2005