A173415 Numbers n such that both the difference and the sum of (n-th prime+1)^2 and (n-th prime)^2 are prime.

1, 3, 10, 128, 201, 223, 246, 309, 357, 393, 424, 482, 526, 815, 887, 909, 1014, 1196, 1543, 1610, 1653, 1674, 1743, 2219, 2302, 2339, 2371, 2475, 2513, 2611, 2948, 3107, 3273, 3419, 3434, 3516, 3555, 3593, 4070, 4203, 4288, 4332, 4389, 4428, 4724, 4793

Offset

1,2

Links

Table of n, a(n) for n=1..46.

Formula

a(n) = Pi(A098717(n)) = A049084(A098717(n)). - R. J. Mathar, Mar 09 2010

Example

a(1)=1 because (1st prime+1)^2 - (1st prime)^2=5 is prime and (1st prime+1)^2 + (1st prime)^2=13 is prime;
a(2)=3 because (3rd prime+1)^2 - (3rd prime)^2=11 is prime and (3rd prime+1)^2 + (3rd prime)^2=61 is prime;
a(3)=10 because (10th prime+1)^2 - (10th prime)^2=59 is prime and (10th prime+1)^2 + (10th prime)^2=1741 is prime;
a(4)=128 because (128th prime+1)^2 - (128th prime)^2=1439 is prime and (128th prime+1)^2 + (128th prime)^2=1035361 is prime.

Mathematica

npsQ[n_]:=Module[{np=Prime[n], a, b}, a=np^2; b=(np+1)^2; And@@PrimeQ[ {a+b, b-a}]]; Select[Range[5000], npsQ] (* Harvey P. Dale, Sep 11 2011 *)

Crossrefs

Cf. A000040, A068501.

Sequence in context: A290059 A062006 A199036 * A199232 A056006 A191363

Adjacent sequences: A173412 A173413 A173414 * A173416 A173417 A173418

Keyword

nonn

Author

Juri-Stepan Gerasimov, Mar 01 2010

Extensions

Extended beyond a(4) by R. J. Mathar, Mar 09 2010

Status

approved