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A262369 A(n,k) is the n-th prime whose decimal expansion begins with the decimal expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

%I A262369 #22 Nov 11 2020 19:08:00
%S A262369 11,2,13,3,23,17,41,31,29,19,5,43,37,211,101,61,53,47,307,223,103,7,
%T A262369 67,59,401,311,227,107,83,71,601,503,409,313,229,109,97,89,73,607,509,
%U A262369 419,317,233,113,101,907,809,79,613,521,421,331,239,127
%N A262369 A(n,k) is the n-th prime whose decimal expansion begins with the decimal expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%H A262369 Alois P. Heinz, <a href="/A262369/b262369.txt">Antidiagonals n = 1..200, flattened</a>
%H A262369 <a href="/index/Pri#piden">Index entries for primes involving decimal expansion of n</a>
%e A262369 Square array A(n,k) begins:
%e A262369 :  11,   2,   3,  41,   5,  61,   7,  83, ...
%e A262369 :  13,  23,  31,  43,  53,  67,  71,  89, ...
%e A262369 :  17,  29,  37,  47,  59, 601,  73, 809, ...
%e A262369 :  19, 211, 307, 401, 503, 607,  79, 811, ...
%e A262369 : 101, 223, 311, 409, 509, 613, 701, 821, ...
%e A262369 : 103, 227, 313, 419, 521, 617, 709, 823, ...
%e A262369 : 107, 229, 317, 421, 523, 619, 719, 827, ...
%e A262369 : 109, 233, 331, 431, 541, 631, 727, 829, ...
%p A262369 u:= (h, t)-> select(isprime, [seq(h*10^t+k, k=0..10^t-1)]):
%p A262369 A:= proc(n, k) local l, p;
%p A262369       l:= proc() [] end; p:= proc() -1 end;
%p A262369       while nops(l(k))<n do p(k):= p(k)+1;
%p A262369         l(k):= [l(k)[], u(k, p(k))[]]
%p A262369       od: l(k)[n]
%p A262369     end:
%p A262369 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A262369 u[h_, t_] := Select[Table[h*10^t + k, {k, 0, 10^t - 1}], PrimeQ];
%t A262369 A[n_, k_] := Module[{l, p}, l[_] = {}; p[_] = -1; While[Length[l[k]] < n, p[k] = p[k]+1; l[k] = Join[l[k], u[k, p[k]]]]; l[k][[n]]];
%t A262369 Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, from Maple *)
%Y A262369 Columns k=1-9 give: A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715.
%Y A262369 Row n=1 gives A018800.
%Y A262369 Main diagonal gives A077345.
%Y A262369 Cf. A077344, A262365.
%K A262369 nonn,base,tabl,look
%O A262369 1,1
%A A262369 _Alois P. Heinz_, Sep 20 2015