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A159351 Smallest prime of the form a^2 + b^2 with 0 < a < b such that a + b = 2n+1. 1

%I

%S 5,13,29,41,61,89,113,149,181,233,269,313,389,421,521,557,613,709,761,

%T 853,929,1013,1109,1201,1301,1409,1553,1637,1741,1861,1997,2113,2269,

%U 2381,2521,2677,2837,2969,3121,3461,3449,3613,3797,4001,4153,4337,4513,4729,5081

%N Smallest prime of the form a^2 + b^2 with 0 < a < b such that a + b = 2n+1.

%C Conjecture: there is always such a prime number.

%C Primes of the form x^2 + y^2 with 0 < x < y such that there are no primes of the form (x+z)^2 + (y-z)^2 for 0 < z < (y-x)/2. Note: a(40) = 3461 > a(41) = 3449, so the order is not maintained. - _Thomas Ordowski_, Jan 21 2017

%D L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999

%D R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994

%D David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005

%e 1^2 + 2^2 = 5 = a(1) = 1.

%e 2^2 + 3^2 = 13 = a(2) < 1^2 + 4^2 = 17.

%e 2^2 + 5^2 = 29 = a(3) < 1^2 + 6^2 = 37.

%e 23^2 + 32^2 = 1553 = a(27) < 1597, 1657, 1693, 1733, 1777, 1877, 1933, 1993, 2273, 2437, 2617, 2713, 2917, 14 prime representations as sum of two squares.

%o (PARI) isok(p, n) = for (i=1, 2*n, if (i^2 + (2*n+1-i)^2 == p, return (1));); 0;

%o a(n) = {my(p = 2); while (! isok(p, n), p = nextprime(p+1)); p;} \\ _Michel Marcus_, Jan 29 2017

%Y Cf. A145354, A159296, A157884, A264904.

%K nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2009

%E A-number in definition and cross-reference corrected, and more terms from _R. J. Mathar_, Apr 24 2009

%E Edited by _Thomas Ordowski_, Jan 25 2017

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Last modified February 28 23:19 EST 2020. Contains 332353 sequences. (Running on oeis4.)