A378459 a(n) is the least k such that the concatenation of 2^n-1 and 2^k-1 is prime, or -1 if there is no such k.

1, 1, 1, 1, 1, 1, 3, 1, 2, 13, 3, 11, 5, 5, 2, 1, 6, 1, 3, 1, 25, 5, 9, 7, 6, 3, 11, 3, 2, 17, 2, 99, 31, 15, 3, 19, 6, 9, 1, 1, 5, 23, 9, 1, 11, 15, 5, 11, 26, 9, 2, 35, 17, 43, 17, 61, 11, 21, 13, 139, 3, 13, 25, 17, 14, 1, 2, 21, 19, 9, 3, 5, 6, 177, 41, 39, 2, 73, 22, 9, 31, 3, 2, 89, 179, 21

Offset

1,7

Comments

a(n) is coprime to n, and is not divisible by 4.

a(1812) > 37000 if it is not -1.

Links

Robert Israel, Table of n, a(n) for n = 1..1811

Example

a(7) = 3 because the concatenation of 2^7-1 = 127 and 2^3-1 = 7 is 1277 which is prime, and neither 1271 nor 1273 is prime.

Maple

tcat:= (a, b) -> 10^(1+ilog10(b))*a+b:
f:= proc(i) local x, j;
    x:= 2^i-1;
    for j from 1 by `if`(i::even, 2, 1) do
     if j mod 4 = 0 or igcd(i, j) > 1 then next fi;
     if isprime(tcat(x, 2^j-1)) then return j fi;
    od
end proc:
map(f, [$1..100]);

Mathematica

idf[a_]:=IntegerDigits[2^a-1]; Table[k=0; Until[PrimeQ[FromDigits[Join[idf[n], idf[k]]]], k++]; k, {n, 86}] (* James C. McMahon, Dec 05 2024 *)

Crossrefs

First column of A378288.

Sequence in context: A254630 A145463 A144107 * A199647 A199149 A198669

Adjacent sequences: A378456 A378457 A378458 * A378460 A378461 A378462

Keyword

nonn,base

Author

Robert Israel, Nov 26 2024

Status

approved