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.

3, 7, 5, 11, 31, 13, 29, 47, 37, 19, 193, 41, 59, 43, 23, 139, 331, 113, 61, 79, 53, 107, 523, 409, 163, 67, 97, 73, 181, 293, 563, 457, 173, 71, 103, 83, 101, 277, 359, 769, 487, 199, 127, 241, 89, 17, 191, 541, 389, 853, 787, 211, 131, 271, 109

Offset

1,1

Links

Alois P. Heinz, Antidiagonals n = 1..100, flattened

Example

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.
Square array A(n,k) begins:
:   3,   7,  11,  29, 193,  139, 107,  181, ...
:   5,  31,  47,  41, 331,  523, 293,  277, ...
:  13,  37,  59, 113, 409,  563, 359,  541, ...
:  19,  43,  61, 163, 457,  769, 389,  937, ...
:  23,  79,  67, 173, 487,  853, 397, 1381, ...
:  53,  97,  71, 199, 787, 1019, 401, 1741, ...
:  73, 103, 127, 211, 829, 1489, 433, 2551, ...
:  83, 241, 131, 251, 991, 1553, 461, 2617, ...

Maple

A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))<n do q:= nextprime(q);
          for h while not isprime((q+h)^2+h)
          do od; p(h):= [p(h)[], q]
        od; p(k)[n]
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

nmax = 12;
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]];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 03 2019 *)

Crossrefs

Column k=1-10 give: A157468, A238664, A238665, A238666, A238667, A238668, A238669, A238670, A238671, A238672.

Rows n=1-10 give: A238673, A238674, A238675, A238676, A238677, A238678, A238679, A238680, A238681, A238682.

Main diagonal gives A238663.

Cf. A238048.

Sequence in context: A074368 A074588 A287865 * A065175 A065283 A352670

Adjacent sequences: A238083 A238084 A238085 * A238087 A238088 A238089

Keyword

nonn,tabl,look

Author

Alois P. Heinz, Feb 17 2014

Status

approved