A275115 Least prime of the form x^2 + n*y^2 with x>0 and y>0.

2, 3, 7, 5, 29, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 173, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 317, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73, 71

Offset

1,1

Comments

Neither x nor y can be zero because the remaining part of the form would then be composite.

a(n) > n.

The differences, d, between a(n) and n are 1, 4, 9, 16, 24, 25, 36, 49, 64, 81, 100, 121, 132, 144, 169, 196, 225, 256, 258, 289, 324, 361, 400, 441, ..., .

Not all 'd's are squares, such as 24, 132, 258, 1032, 1167, 1518, 2103, 2472, 2652, 2706, 5793. It is conjectured that this list is complete.

d=1 for A006093;

d=4 for A172367;

d=9 for n: 8, 14, 20, 32, 34, 38, 44, 50, 62, 64, 74, 80, 92, 94, 98, 104, 118, 122, 128, 140, 142, 154, 158, ..., ;

d=16 for n: 21, 31, 45, 51, 73, 81, 87, 91, 111, 115, 121, 141, 151, 157, 165, 181, 183, 211, 213, 217, 241, ..., ;

d=25 for n: 48, 54, 76, 84, 114, 124, 132, 168, 174, 186, 204, 208, 216, 244, 246, 252, 258, 286, 288, 324, ..., ;

d=36 for n: 11, 17, 23, 35, 47, 53, 61, 65, 71, 77, 95, 101, 113, 131, 137, 143, 155, 161, 191, 197, 203, 205, ..., ;

d=49 for n: 24, 90, 144, 234, 264, 300, 318, 360, 390, 450, 472, 492, 528, 550, 558, 564, 624, 670, 678, 712, ..., ;

and for the nonsquare differences of 24, 132, 258, 1032, 1167, 1518, 2103, 2472, 2652, 2706 and 5793l, their n's are 5, 41, 59, 341, 314, 479, 626, 749, 881, 755 and 1784, respectively.

Least n that has as its difference k^2: 1, 3, 8, 21, 48, 11, 24, 117, 26, 139, 120, 29, 294, 201, 134, 621, 468, 179, 792, 1269, 356, 1249, 754, 251, 696, ..., .

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Zak Seidov, First ten primes of the form x^2+n*y^2 with x>=0, y>=0, n=1..1000.

Formula

a(n-1) = n iff n is prime.

Example

a(1) = 2 since it equals 1^2+1*1^2;
a(2) = 3 since it equals 1^2+2*1^2;
a(3) = 7 since it equals 2^2+3*1^2;
a(4) = 5 since it equals 1^2+4*1^2;
a(5) = 29 since it equals 3^2+5*2^2; etc.

Mathematica

f[n_] := Block[{p = NextPrime@ n, y}, While[y = 1; While[p > n*y^2 && !IntegerQ[ Sqrt[p - n*y^2]], y++]; !IntegerQ[ Sqrt[p - n*y^2]], p = NextPrime@ p]; p]; Array[f, 70]

Prog

(PARI) a(n)=if(n==1, return(2)); my(best, x=1+n%2, t); while(!isprime(best=x^2+n), x += 2); for(y=2, sqrtint((best-2)\n), t=best-n*y^2; if(t<1, return(best)); for(x=1, sqrtint(t), if(isprime(t=x^2+n*y^2) && t<best, best=t))); best \\ Charles R Greathouse IV, Jul 17 2016

Crossrefs

Cf. A002350, A232174, A212602, A212603, A212604, A212605, A244030, A244031, A006093, A172367.

Sequence in context: A351494 A155766 A153488 * A085399 A063696 A258126

Adjacent sequences: A275112 A275113 A275114 * A275116 A275117 A275118

Keyword

nonn

Author

Zak Seidov and Robert G. Wilson v, Jul 17 2016

Status

approved