A378288 Array read by antidiagonals: row k consists of the positive integers j for which the concatenation of 2^k - 1 and 2^j - 1 is prime.

1, 2, 1, 3, 3, 1, 5, 5, 2, 1, 6, 9, 7, 3, 1, 9, 15, 10, 5, 2, 1, 13, 27, 11, 9, 3, 7, 3, 18, 45, 13, 15, 6, 17, 6, 1, 19, 59, 14, 45, 9, 41, 13, 3, 2, 26, 211, 25, 61, 11, 101, 125, 7, 5, 13, 43, 303, 62, 65, 13, 157, 150, 9, 11, 27, 3, 46, 425, 70, 227, 23, 367, 195, 11, 14, 43, 14, 11

Offset: 1

Author

Robert Israel, Nov 26 2024

Keywords

• nonn
• base
• tabl

Comments

No terms are divisible by 4.

All terms in row k are coprime to k.

Conjecture: all rows have infinitely many terms, and all positive integers not divisible by 4 appear in infinitely many rows.

k - 1 is in row k iff k is in A301806.

Examples

			The array starts
   1    2    3    5    6    9   13   18 ...
   1    3    5    9   15   27   45   59 ...
   1    2    7   10   11   13   14   25 ...
   1    3    5    9   15   45   61   65 ...
   1    2    3    6    9   11   13   23 ...
   1    7   17   41  101  157  367  571 ...
   3    6   13  125  150  195  634 1282 ...
   1    3    7    9   11   23   27   39 ...
a(3,4) = 10 is a term in row 3 because the concatenation of 2^3 - 1 = 7 and 2^10 - 1 = 1023 is 71023, which is prime.
		

Crossrefs

Cf. A000225, A301806.

Programs

Maple

tcat:= (a,b) -> 10^(1+ilog10(b))*a+b:
N:= 8: # for the top left N x N array
M:= Matrix(N, N):
for i from 1 to N do
   count:= 0:
   x:= 2^i-1;
   for j from 1 by `if`(i::even,2,1) while count + i < N do
     if j mod 4 = 0 or igcd(i,j) > 1 then next fi;
     if isprime(tcat(x,2^j-1)) then count:= count+1; M[i,count]:= j fi;
   od;
od:
M;
seq(seq(M[k,1+d-k], k=1..d), d=1..N-1);