

A159351


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


1



5, 13, 29, 41, 61, 89, 113, 149, 181, 233, 269, 313, 389, 421, 521, 557, 613, 709, 761, 853, 929, 1013, 1109, 1201, 1301, 1409, 1553, 1637, 1741, 1861, 1997, 2113, 2269, 2381, 2521, 2677, 2837, 2969, 3121, 3461, 3449, 3613, 3797, 4001, 4153, 4337, 4513, 4729, 5081
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OFFSET

1,1


COMMENTS

Conjecture: there is always such a prime number.
Primes of the form x^2 + y^2 with 0 < x < y such that there are no primes of the form (x+z)^2 + (yz)^2 for 0 < z < (yx)/2. Note: a(40) = 3461 > a(41) = 3449, so the order is not maintained.  Thomas Ordowski, Jan 21 2017


REFERENCES

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: SpringerVerlag, 1994
David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005


LINKS

Table of n, a(n) for n=1..49.


EXAMPLE

1^2 + 2^2 = 5 = a(1) = 1.
2^2 + 3^2 = 13 = a(2) < 1^2 + 4^2 = 17.
2^2 + 5^2 = 29 = a(3) < 1^2 + 6^2 = 37.
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.


PROG

(PARI) isok(p, n) = for (i=1, 2*n, if (i^2 + (2*n+1i)^2 == p, return (1)); ); 0;
a(n) = {my(p = 2); while (! isok(p, n), p = nextprime(p+1)); p; } \\ Michel Marcus, Jan 29 2017


CROSSREFS

Cf. A145354, A159296, A157884, A264904.
Sequence in context: A158756 A185086 A277701 * A163251 A247177 A146286
Adjacent sequences: A159348 A159349 A159350 * A159352 A159353 A159354


KEYWORD

nonn


AUTHOR

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


EXTENSIONS

Anumber in definition and crossreference corrected, and more terms from R. J. Mathar, Apr 24 2009
Edited by Thomas Ordowski, Jan 25 2017


STATUS

approved



