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%I #32 May 03 2019 11:53:21
%S 3,7,5,11,31,13,29,47,37,19,193,41,59,43,23,139,331,113,61,79,53,107,
%T 523,409,163,67,97,73,181,293,563,457,173,71,103,83,101,277,359,769,
%U 487,199,127,241,89,17,191,541,389,853,787,211,131,271,109
%N Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k is the increasing list of all primes p such that (p+k)^2+k is prime but (p+j)^2+j is not prime for all 0<j<k.
%H Alois P. Heinz, <a href="/A238086/b238086.txt">Antidiagonals n = 1..100, flattened</a>
%e Column k=3 contains prime 47 because (47+3)^2+3 = 2503 is prime and (47+2)^2+2 = 2403 = 3^3*89 and (47+1)^2+1 = 2305 = 5*461 are composite.
%e Square array A(n,k) begins:
%e : 3, 7, 11, 29, 193, 139, 107, 181, ...
%e : 5, 31, 47, 41, 331, 523, 293, 277, ...
%e : 13, 37, 59, 113, 409, 563, 359, 541, ...
%e : 19, 43, 61, 163, 457, 769, 389, 937, ...
%e : 23, 79, 67, 173, 487, 853, 397, 1381, ...
%e : 53, 97, 71, 199, 787, 1019, 401, 1741, ...
%e : 73, 103, 127, 211, 829, 1489, 433, 2551, ...
%e : 83, 241, 131, 251, 991, 1553, 461, 2617, ...
%p A:= proc() local h, p, q; p, q:= proc() [] end, 2;
%p proc(n, k)
%p while nops(p(k))<n do q:= nextprime(q);
%p for h while not isprime((q+h)^2+h)
%p do od; p(h):= [p(h)[], q]
%p od; p(k)[n]
%p end
%p end():
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t nmax = 12;
%t col[k_] := col[k] = Reap[For[cnt = 0; p = 2, cnt < nmax, p = NextPrime[p], If[PrimeQ[(p+k)^2+k] && AllTrue[Range[k-1], !PrimeQ[(p+#)^2+#]&], cnt++; Sow[p]]]][[2, 1]];
%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_, May 03 2019 *)
%Y Column k=1-10 give: A157468, A238664, A238665, A238666, A238667, A238668, A238669, A238670, A238671, A238672.
%Y Rows n=1-10 give: A238673, A238674, A238675, A238676, A238677, A238678, A238679, A238680, A238681, A238682.
%Y Main diagonal gives A238663.
%Y Cf. A238048.
%K nonn,tabl,look
%O 1,1
%A _Alois P. Heinz_, Feb 17 2014
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