A175349 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.

4, 6, 40, 39, 43, 108, 113, 79, 368, 466, 500, 149, 361, 344, 377, 53, 59, 988, 542, 2272, 2121, 1103, 2259, 356, 609, 1253, 3304, 3434, 876, 2929, 4078, 387, 393, 2226, 4787, 1687, 630, 2615, 1336, 5561, 2874, 5820, 382, 4033, 12608, 8391, 13506, 14276, 8931, 14662

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

Maple

nextcomp:= proc(c)
  if isprime(c+1) then c+2 else c+1 fi
end proc:
f:= proc(p, c)
   local Bp, dp, Bc, dc, m, flag1, flag2, x1, x2;
   Bp:= convert(convert(p, binary), string); dp:= length(Bp);
   Bc:= convert(convert(c, binary), string); dc:= length(Bc);
   if dc < dp and StringTools:-Search(Bc, Bp) <> 0 then return p
   elif dp < dc and StringTools:-Search(Bp, Bc) <> 0 then return c
   fi;
   for m from min(dp, dc) to 1 by -1 do
     flag1:= Bp[1..m] = Bc[-m..-1];
     flag2:= Bc[1..m] = Bp[-m..-1];
     if flag1 then x1:= 2^(dp-m) * c + (p mod 2^(dp-m)) fi;
     if flag2 then x2:= 2^(dc-m) * p + (c mod 2^(dc-m)) fi;
     if flag1 and flag2 then return min(x1, x2)
     elif flag1 then return x1
     elif flag2 then return x2
     fi;
   od;
   min(p*2^dc + c, c*2^dp+p)
end proc:
p:= 1: c:= 2: R:= NULL:
for n from 1 to 100 do
  p:= nextprime(p); c:= nextcomp(c);
  R:= R, f(p, c)
od:
R; # Robert Israel, Nov 28 2024

Mathematica

comp[n_] := FixedPoint[n + 1 + PrimePi[#] &, n + 1 + PrimePi[n]]; sub[n_, x_] := MemberQ[Partition[IntegerDigits[n, 2], IntegerLength[x, 2], 1],
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 *)

Crossrefs

Cf. A004676, A115454, A378479.

Sequence in context: A103234 A074061 A125081 * A281223 A023644 A319672

Adjacent sequences: A175346 A175347 A175348 * A175350 A175351 A175352

Keyword

base,nonn,look

Author

Leroy Quet, Apr 19 2010

Extensions

a(9)-a(50) from Giovanni Resta, Jul 02 2018

Status

approved