%I #12 Jul 02 2018 10:13:40
%S 4,6,40,39,43,108,113,79,368,466,500,149,361,344,377,53,59,988,542,
%T 2272,2121,1103,2259,356,609,1253,3304,3434,876,2929,4078,387,393,
%U 2226,4787,1687,630,2615,1336,5561,2874,5820,382,4033,12608,8391,13506,14276,8931,14662
%N a(n) is the smallest positive integer that, when written in binary, contains the binary representations of both the n-th prime and the n-th composite as (possibly overlapping) substrings.
%e The 7th prime is 17, which is 10001 in binary. The 7th composite is 14, which is 1110 in binary. The smallest positive integer that, when written in binary, contains these binary representations as substrings is 113, which is 1110001 in binary. a(7) = 113, therefore.
%t comp[n_] := FixedPoint[n + 1 + PrimePi[#] &, n + 1 + PrimePi[n]]; sub[n_, x_] := MemberQ[Partition[IntegerDigits[n, 2], IntegerLength[x, 2], 1],
%t IntegerDigits[x, 2]]; a[n_] := Block[{c = comp[n], p = Prime[n], k}, k = Max[p, c]; While[! sub[k,p] || ! sub[k,c], k++]; k]; Array[a, 50] (* _Giovanni Resta_, Jul 02 2018 *)
%Y Cf. A004676, A115454.
%K base,nonn
%O 1,1
%A _Leroy Quet_, Apr 19 2010
%E a(9)-a(50) from _Giovanni Resta_, Jul 02 2018
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