A378479 Numbers k such that in base 2 the k-th composite is a substring of the k-th prime.

16, 17, 98, 210, 654, 3386, 3387, 3388, 3389, 3392, 3395, 3397, 3398, 3401, 3504, 4806, 22401, 27997, 30930, 75126, 109303, 119466, 119467, 221344, 265167, 391691, 412566, 772432, 949072, 1451888, 2456497, 2739020, 2963199, 4942623, 4942624, 4942631, 4942632, 4942634, 4942636, 4942637, 4942638

Offset: 1

Author

Robert Israel, Nov 28 2024

Keywords

• nonn
• base

Comments

Numbers k such that A175349(k) = prime(k).

Examples

			a(3) = 98 is a term because the 98th composite is 130 which is 10000010 in binary, the 98th prime is 521 which is 1000001001 in binary, and the first 8 bits of 1000001001 are 10000010.
		

Crossrefs

Cf. A000040, A002808, A175349.

Programs

Maple

g:= proc(p,c)
   StringTools:-Search(convert(convert(c,binary),string),
                         convert(convert(p,binary),string)) <> 0
end proc:
nextcomp:= proc(c)
  if isprime(c+1) then c+2 else c+1 fi
end proc:
p:= 1: c:= 2: Res:= NULL: count:= 0:
for n from 1 to 10^7 do
  p:= nextprime(p);
  c:= nextcomp(c);
  if g(p,c) then Res:= Res, n; count:= count+1; fi
od:
R;

Mathematica

p = 1;
c = 2;
Table[
 p = NextPrime[p];
 c += If[PrimeQ[c + 1], 2, 1];
 mod = 2^BitLength[c];
 test = NestWhile[BitShiftRight, p,
   BitAnd[#, mod - 1] != c && # > c &];
 If[test >= c, n, Nothing]
 , {n, 10^5}]
(* David Trimas, Dec 01 2024 *)