A335410 Primes prime(k) such that 2*(prime(k)^2 - prime(k-1)^2) is a perfect square.

11, 19, 73, 83, 227, 443, 883, 1091, 1153, 1931, 2593, 2609, 3529, 4051, 7451, 13691, 15139, 16649, 20809, 26921, 34849, 45377, 46819, 53147, 56171, 69193, 74507, 74531, 83233, 91811, 95483, 103067, 103969, 106937, 110459, 112339, 149059, 149771, 176419, 180001

Offset

1,1

Comments

2*(prime(n)^2 - prime(n-1)^2) represents the integer coefficient of the difference in areas between the two circles passing through the origin with centers located at (prime(n), prime(n)) and (prime(n-1), prime(n-1)).

Among the first 200000 primes 2593 and 2609 are the only consecutive primes in this sequence.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

Prime(5) = 11, prime(4) = 7, 2*(11^2 - 7^2) = 12^2, so 11 is in the sequence.
Prime(559) = 4051, prime(558) = 4049, 2*(4051^2 - 4049^2) = 180^2, so 4051 is in the sequence.

Mathematica

Select[Prime@ Range[2, 17000], IntegerQ@ Sqrt[2 (#^2 - NextPrime[#, -1]^2)] &] (* Giovanni Resta, Jun 06 2020 *)

Prog

(PARI) lista(nn) = {my(pp=2); forprime (p=3, nn, if (issquare(2*(p^2 - pp^2)), print1(p, ", ")); pp = p; ); } \\ Michel Marcus, Jun 25 2020

Crossrefs

Cf. A001248, A332615.

Sequence in context: A290078 A089032 A344042 * A106231 A107637 A229542

Adjacent sequences: A335407 A335408 A335409 * A335411 A335412 A335413

Keyword

nonn

Author

Jeff Brown, Jun 06 2020

Extensions

More terms from Giovanni Resta, Jun 06 2020

Status

approved