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%I #9 Mar 25 2017 20:32:01
%S 2,3,4,5,5,7,10,10,7,9,7,5,13,15,15,15,17,17,15,17,18,17,29,21,17,28,
%T 20,31,21,36,15,15,35,42,41,35,33,22,45,22,36,46,50,41,37,33,21,40,38,
%U 31,66,33,60,45,51,42,42,60,35,38,63,65,75,75,39,71,36,75,67,74,67,68,78
%N Least number whose binary representation is not contained in that of n!.
%C a(A093710(n)) > n, a(A093711(n)) <= n.
%C Sum_{n=1..3000} a(n) = 4179226. The average value of a(n)/n -> 0.93+ and except for some initial terms, < 212, 0.90 < a(n)/n < 0.95. But there is graphical evidence that this value may increase. - _Robert G. Wilson v_, Apr 21 2004
%C a(n) = n only for 5, 13, 15, 17, 513, 517, 1041, 1538, 2138, n <= 2500. a(n) = n+1 for n=1, 2, 3, 4, 6, 14, 134, 137, 155, 169, 216, 313, 530, 1035 & 2402. - _Robert G. Wilson v_, Apr 21 2004
%C a(n) >= 3n/2 for n=1, 2, 334, 335, 2693 & 2739. - _Robert G. Wilson v_, Apr 21 2004
%C a(n) <= n/2 for n=12, 31, 32, 47, 122, 142, 152, 188, 303, 378, 443, 548, 598, 1319, 1354, 1420, 2127, 2137, 2223, 2230, 2368, 2433, 2571. - _Robert G. Wilson v_, Apr 21 2004
%e a(7)=10 because 7! = 1001110110000_b and 1_b, 10_b, 11_b, 100_b, 101_b, 110_b, 111_b, 1000_b & 1001_b are substrings, but 10_d = 1010_b is not a substring. - _Robert G. Wilson v_, Apr 21 2004
%t f[n_] := ToString[ FromDigits[ IntegerDigits[n, 2]]]; g[n_] := g[n] = Block[{k = 1, s = f[n! ]}, While[ StringPosition[ s, f[k]] != {}, k++ ]; k]; Table[ g[n], {n, 75}] (* _Robert G. Wilson v_, Apr 21 2004 *)
%Y Cf. A036603, A007088, A000142, A092601.
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, Apr 13 2004
%E Edited by _N. J. A. Sloane_, Sep 15 2008 at the suggestion of _R. J. Mathar_
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