OFFSET
1,2
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.
EXAMPLE
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.
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);
CROSSREFS
KEYWORD
AUTHOR
Robert Israel, Nov 26 2024
STATUS
approved