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%I #15 Jul 21 2023 04:28:04
%S 7,19,41,11,23,43,71,47,17,29,113,157,53,23,83,163,59,167,347,173,353,
%T 47,67,131,227,431,41,53,73,101,137,181,233,293,521,613,617,47,59,79,
%U 107,239,367,443,619,719,827,83,191,947,89,197,953,383,97,257,317,461,1217,71
%N Irregular triangle T(n, k) = prime(n+1) + (2*k)^2 if ( prime(n+1) + (2*k)^2 ) is prime, for 1 <= k <= n+1, n >= 1, flattened.
%H G. C. Greubel, <a href="/A120720/b120720.txt">Table of n, a(n) for n = 1..5000</a>
%F T(n, k) = prime(n+1) + (2*k)^2 if ( prime(n+1) + (2*k)^2 ) is prime, for 1 <= k <= n+1, n >= 1.
%e Triangle formed from prime(n+1) + (2*k)^2:
%e 7, 19;
%e 9, 21, 41;
%e 11, 23, 43, 71;
%e 15, 27, 47, 75, 111;
%e 17, 29, 49, 77, 113, 157;
%e 21, 33, 53, 81, 117, 161, 213;
%e 23, 35, 55, 83, 119, 163, 215, 275;
%e 27, 39, 59, 87, 123, 167, 219, 279, 347;
%e keeping only the prime values gives T(n,k):
%e 7, 19;
%e 41;
%e 11, 23, 43, 71;
%e 47;
%e 17, 29, 113, 157;
%e 53;
%e 23, 83, 163;
%e 59, 167, 347;
%t T[n_, k_]:= If[PrimeQ[(2*k)^2 + Prime[n+1]], (2*k)^2 + Prime[n+1], {}];
%t Table[T[n, k], {n,15}, {k,n+1}]//Flatten
%o (Magma) [NthPrime(n+1) + 4*k^2: k in [1..n+1], n in [1..21] | IsPrime(NthPrime(n+1) + 4*k^2) ]; // _G. C. Greubel_, Jul 20 2023
%o (SageMath)
%o def A120720(n,k): return nth_prime(n+1) + 4*k^2 if is_prime(nth_prime(n+1) + 4*k^2) else []
%o flatten([[A120720(n,k) for k in range(1,n+2)] for n in range(1,22)]) # _G. C. Greubel_, Jul 20 2023
%K nonn,tabf
%O 1,1
%A _Roger L. Bagula_, Aug 15 2006
%E Edited by _G. C. Greubel_, Jul 20 2023
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