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A238086 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

%I A238086 #32 May 03 2019 11:53:21
%S A238086 3,7,5,11,31,13,29,47,37,19,193,41,59,43,23,139,331,113,61,79,53,107,
%T A238086 523,409,163,67,97,73,181,293,563,457,173,71,103,83,101,277,359,769,
%U A238086 487,199,127,241,89,17,191,541,389,853,787,211,131,271,109
%N A238086 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 A238086 Alois P. Heinz, <a href="/A238086/b238086.txt">Antidiagonals n = 1..100, flattened</a>
%e A238086 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 A238086 Square array A(n,k) begins:
%e A238086 :   3,   7,  11,  29, 193,  139, 107,  181, ...
%e A238086 :   5,  31,  47,  41, 331,  523, 293,  277, ...
%e A238086 :  13,  37,  59, 113, 409,  563, 359,  541, ...
%e A238086 :  19,  43,  61, 163, 457,  769, 389,  937, ...
%e A238086 :  23,  79,  67, 173, 487,  853, 397, 1381, ...
%e A238086 :  53,  97,  71, 199, 787, 1019, 401, 1741, ...
%e A238086 :  73, 103, 127, 211, 829, 1489, 433, 2551, ...
%e A238086 :  83, 241, 131, 251, 991, 1553, 461, 2617, ...
%p A238086 A:= proc() local h, p, q; p, q:= proc() [] end, 2;
%p A238086       proc(n, k)
%p A238086         while nops(p(k))<n do q:= nextprime(q);
%p A238086           for h while not isprime((q+h)^2+h)
%p A238086           do od; p(h):= [p(h)[], q]
%p A238086         od; p(k)[n]
%p A238086       end
%p A238086     end():
%p A238086 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A238086 nmax = 12;
%t A238086 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 A238086 A[n_, k_] := col[k][[n]];
%t A238086 Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, May 03 2019 *)
%Y A238086 Column k=1-10 give: A157468, A238664, A238665, A238666, A238667, A238668, A238669, A238670, A238671, A238672.
%Y A238086 Rows n=1-10 give: A238673, A238674, A238675, A238676, A238677, A238678, A238679, A238680, A238681, A238682.
%Y A238086 Main diagonal gives A238663.
%Y A238086 Cf. A238048.
%K A238086 nonn,tabl,look
%O A238086 1,1
%A A238086 _Alois P. Heinz_, Feb 17 2014