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A262365 A(n,k) is the n-th prime whose binary expansion begins with the binary expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

%I A262365 #20 Oct 25 2021 06:19:25
%S A262365 2,2,3,3,5,5,17,7,11,7,5,19,13,17,11,13,11,37,29,19,13,7,53,23,67,31,
%T A262365 23,17,17,29,97,41,71,53,37,19,19,67,31,101,43,73,59,41,23,41,37,71,
%U A262365 59,103,47,79,61,43,29,11,43,73,131,61,107,83,131,97,47,31
%N A262365 A(n,k) is the n-th prime whose binary expansion begins with the binary expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%H A262365 Alois P. Heinz, <a href="/A262365/b262365.txt">Antidiagonals n = 1..200, flattened</a>
%H A262365 <a href="/index/Pri#piden">Index entries for primes involving decimal expansion of n</a>
%e A262365 Square array A(n,k) begins:
%e A262365 :  2,  2,  3,  17,  5,  13,   7,  17, ...
%e A262365 :  3,  5,  7,  19, 11,  53,  29,  67, ...
%e A262365 :  5, 11, 13,  37, 23,  97,  31,  71, ...
%e A262365 :  7, 17, 29,  67, 41, 101,  59, 131, ...
%e A262365 : 11, 19, 31,  71, 43, 103,  61, 137, ...
%e A262365 : 13, 23, 53,  73, 47, 107, 113, 139, ...
%e A262365 : 17, 37, 59,  79, 83, 109, 127, 257, ...
%e A262365 : 19, 41, 61, 131, 89, 193, 227, 263, ...
%p A262365 u:= (h, t)-> select(isprime, [seq(h*2^t+k, k=0..2^t-1)]):
%p A262365 A:= proc(n, k) local l, p;
%p A262365       l:= proc() [] end; p:= proc() -1 end;
%p A262365       while nops(l(k))<n do p(k):= p(k)+1;
%p A262365         l(k):= [l(k)[], u(k,p(k))[]]
%p A262365       od: l(k)[n]
%p A262365     end:
%p A262365 seq(seq(A(n,1+d-n), n=1..d), d=1..14);
%t A262365 nmax = 14;
%t A262365 col[k_] := col[k] = Module[{bk = IntegerDigits[k, 2], lk, pp = {}, coe = 1}, lbk = Length[bk]; While[Length[pp] < nmax, pp = Select[Prime[Range[ coe*nmax]], Quiet@Take[IntegerDigits[#, 2], lbk] == bk&]; coe++]; pp];
%t A262365 A[n_, k_] := col[k][[n]];
%t A262365 Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 25 2021 *)
%Y A262365 Columns k=1-7 give: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
%Y A262365 Row n=1 gives A164022.
%Y A262365 Main diagonal gives A262366.
%Y A262365 Cf. A262350, A262369.
%K A262365 nonn,look,base,tabl
%O A262365 1,1
%A A262365 _Alois P. Heinz_, Sep 20 2015