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A264904 Primes of the form x^2 + y^2 with 0 < x < y such that all the numbers (x-a)^2 + (y+a)^2 (a = 1,...,x) are composite. 2
5, 17, 37, 53, 101, 109, 197, 257, 293, 401, 409, 577, 677, 701, 733, 857, 1093, 1297, 1373, 1601, 1609, 1697, 2029, 2141, 2213, 2417, 2917, 3137, 3253, 3373, 3389, 3853, 4261, 4357, 4493, 4909, 5209, 5477, 5641, 5801, 6257, 7057, 7229, 7573, 7937, 8101, 8837, 9029, 9413, 9613, 10009, 10429, 10453, 10613, 12101, 12109, 12553, 13457, 13693, 14177 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Note that the sequence contains all primes of the form n^2 + 1 with n > 1. A conjecture of Landau states that there are infinitely many primes of the form n^2 + 1.

Conjecture: For any prime p > 5 of the form x^2 + y^2 (0 < x < y), there is a prime q not equal to p of the form u^2 + v^2 (0 < u < v) with u + v = x + y.

A subsequence of A002313. - Altug Alkan, Dec 18 2015

Conjecture: each odd number m > 1 is a unique sum m = x + y with 0 < x < y, where x^2 + y^2 is in the sequence. - Thomas Ordowski, Jan 16 2017

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3500

EXAMPLE

a(1) = 5 since 5 = 1^2 + 2^2 is a prime with 0 < 1 < 2, and 0^2 + 3^2 = 9 is composite.

a(4) = 53 since 53 = 2^2 + 7^2 is a prime with 0 < 2 < 7, and 0^2 + 9^2 = 81 and 1^2 + 8^2 = 65 are both composite.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

Y[n_]:=Y[n]=Sum[If[SQ[n-4*y^2], 2y, 0], {y, 0, Sqrt[n/4]}]

X[n_]:=X[n]=Sqrt[n-Y[n]^2]

p[n_]:=p[n]=Prime[n]

x[n_]:=x[n]=X[p[n]]

y[n_]:=y[n]=Y[p[n]]

n=0; Do[If[Mod[p[k]-1, 4]==0, Do[If[PrimeQ[a^2+(x[k]+y[k]-a)^2], Goto[aa]], {a, 0, Min[x[k], y[k]]-1}]; n=n+1; Print[n, " ", p[k]]]; Label[aa]; Continue, {k, 2, 1669}]

CROSSREFS

Cf. A000040, A000290, A002144, A002313, A002496, A264865, A264866.

Sequence in context: A080167 A060245 A119456 * A257582 A273538 A273212

Adjacent sequences:  A264901 A264902 A264903 * A264905 A264906 A264907

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 28 2015

STATUS

approved

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Last modified February 26 21:29 EST 2020. Contains 332295 sequences. (Running on oeis4.)