A262365 - OEIS

%I #20 Oct 25 2021 06:19:25

%S 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 23,17,17,29,97,41,71,53,37,19,19,67,31,101,43,73,59,41,23,41,37,71,

%U 59,103,47,79,61,43,29,11,43,73,131,61,107,83,131,97,47,31

%N 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 Alois P. Heinz, <a href="/A262365/b262365.txt">Antidiagonals n = 1..200, flattened</a>

%H <a href="/index/Pri#piden">Index entries for primes involving decimal expansion of n</a>

%e Square array A(n,k) begins:

%e :  2,  2,  3,  17,  5,  13,   7,  17, ...

%e :  3,  5,  7,  19, 11,  53,  29,  67, ...

%e :  5, 11, 13,  37, 23,  97,  31,  71, ...

%e :  7, 17, 29,  67, 41, 101,  59, 131, ...

%e : 11, 19, 31,  71, 43, 103,  61, 137, ...

%e : 13, 23, 53,  73, 47, 107, 113, 139, ...

%e : 17, 37, 59,  79, 83, 109, 127, 257, ...

%e : 19, 41, 61, 131, 89, 193, 227, 263, ...

%p u:= (h, t)-> select(isprime, [seq(h*2^t+k, k=0..2^t-1)]):

%p A:= proc(n, k) local l, p;

%p       l:= proc() [] end; p:= proc() -1 end;

%p       while nops(l(k))<n do p(k):= p(k)+1;

%p         l(k):= [l(k)[], u(k,p(k))[]]

%p       od: l(k)[n]

%p     end:

%p seq(seq(A(n,1+d-n), n=1..d), d=1..14);

%t nmax = 14;

%t 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 A[n_, k_] := col[k][[n]];

%t Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 25 2021 *)

%Y Columns k=1-7 give: A000040, A080165, A080166, A262286, A262284, A262287, A262285.

%Y Row n=1 gives A164022.

%Y Main diagonal gives A262366.

%Y Cf. A262350, A262369.

%K nonn,look,base,tabl

%O 1,1

%A _Alois P. Heinz_, Sep 20 2015