A244030 -id:A244030 - cp's OEIS frontend

A244031 Integers n>1 such that the quadratic form x^2+n*y^2 does not represent a prime strictly between n and 2n.

3, 5, 8, 11, 17, 23, 24, 26, 29, 35, 41, 56, 59, 68, 83, 89, 107, 119, 120, 125, 134, 179, 185, 194, 206, 251, 263, 269, 290, 293, 314, 326, 341, 356, 371, 389, 401, 404, 461, 464, 479, 489, 491, 524, 545, 569, 593, 626

Offset: 1

N. J. A. Sloane, Jun 22 2014

nonn

Or: Positive numbers n such that n + k^2 is composite for all 1 <= k^2 <= n.

The next term a(105), if it exists, is > 156*10^6. - M. F. Hasler, May 07 2018

N. J. A. Sloane, Table of n, a(n) for n = 1..104 (cf. Jagy & Kaplansky).
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]

Union of A244029 (subsequence of primes) and A244030 (composite terms).

is(n)=!for(k=1,sqrtint(n),isprime(n+k^2)&&return) \\ M. F. Hasler, May 07 2018

Added "n>1" as suggested by David J. Seal. - N. J. A. Sloane, May 19 2018

A244029 Primes p such that the quadratic form x^2+p*y^2 does not represent a prime strictly between p and 2p.

Original entry on oeis.org

3, 5, 11, 17, 23, 29, 41, 59, 83, 89, 107, 179, 251, 263, 269, 293, 389, 401, 461, 479, 491, 569, 593, 881, 929, 1319, 1619, 1931, 2531, 2789, 3461, 3701, 4919, 5309, 7589, 9749, 26171

Offset: 1

N. J. A. Sloane, Jun 22 2014

nonn

William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]

Cf. A244030, A244031.

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

Original entry on oeis.org

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

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

nonn

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, ..., .

			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.

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.

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

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]

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) && tCharles R Greathouse IV, Jul 17 2016

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

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A244031 Integers n>1 such that the quadratic form x^2+n*y^2 does not represent a prime strictly between n and 2n.

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A244029 Primes p such that the quadratic form x^2+p*y^2 does not represent a prime strictly between p and 2p.

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

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