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
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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 1..100, flattened
Crossrefs
Programs
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Maple
A:= proc() local h, p, q; p, q:= proc() [] end, 2; proc(n, k) while nops(p(k)) -
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 *)