Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #28 Jan 30 2017 14:06:45
%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