A217503 Squared distance between consecutive primes of the form 4k+1 (see below).

1, 2, 2, 2, 2, 10, 8, 10, 8, 4, 2, 10, 4, 20, 18, 10, 2, 20, 58, 8, 40, 2, 40, 20, 10, 90, 2, 20, 10, 116, 2, 8, 20, 10, 2, 10, 20, 26, 4, 146, 8, 34, 10, 40, 34, 40, 2, 20, 2, 160, 50, 10, 180, 2, 180, 90, 58, 40, 130, 16, 116, 194, 50, 136, 74, 34, 52, 40

Offset

1,2

Comments

Every prime p of the form 4k+1 has a unique solution p = x^2 + y^2. This sequence gives the squared distance between points (x,y) for consecutive primes of this form.

The squares mutual distance consecutive points with coordinates x(n) = A002331(n) and y(n) = A002330(n), where x(n)^2 + y(n)^2 = A002313(n) is prime.

Theorem: a(n) =/= A082073(n-1) for all n > 1. Generally, it can be shown that there is no pair of primes p = a^2 + b^2 and q = x^2 + y^2 such that (a - x)^2 + (b - y)^2 = |p - q| > 0.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Example

5 = 1^2 + 2^2 and 13 = 2^2 + 3^2. The squared distance between the points (1,2) and (2,3) is 2, the second term of this sequence.

Mathematica

nn = 200; p = Select[Prime[Range[nn]], Mod[#, 4] == 1 &]; q = {1, 1}; Table[pp = PowersRepresentations[p[[i]], 2, 2][[1]]; d = pp - q; q = pp; d[[1]]^2 + d[[2]]^2, {i, Length[p] - 1}] (* T. D. Noe, Oct 19 2012 *)

Crossrefs

Cf. A002313, A002330, A002331.

Sequence in context: A265447 A156538 A249768 * A165466 A175392 A289910

Adjacent sequences: A217500 A217501 A217502 * A217504 A217505 A217506

Keyword

nonn

Author

Thomas Ordowski, Oct 05 2012

Status

approved