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 A217503 Squared distance between consecutive primes of the form 4k+1 (see below). 2
 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 (list; graph; refs; listen; history; text; internal format)
 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][]; d = pp - q; q = pp; d[]^2 + d[]^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

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Last modified June 6 05:06 EDT 2023. Contains 363139 sequences. (Running on oeis4.)