A000926 -id:A000926 - cp's OEIS frontend

A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

1, 2, 1, 2, 3, 1, 3, 7, 1, 5, 9, 13, 1, 5, 9, 1, 7, 1, 7, 9, 11, 15, 23, 25, 1, 9, 17, 25, 1, 13, 25, 1, 9, 11, 19, 1, 13, 25, 37, 1, 9, 13, 17, 25, 29, 49, 1, 19, 31, 49, 1, 9, 17, 25, 33, 41, 49, 57, 1, 19, 25, 43, 49, 67, 1, 25, 37, 1, 9, 15, 23, 25, 31, 47, 49, 71, 81, 1, 25, 49, 73, 1, 9

Offset: 1

T. D. Noe, Apr 28 2008

Each row begins with 1. For example, the 12th row is for N=13. The numbers in that row are 1, 9, 17, 25, 29 and 49, which means that the primes represented by the quadratic form x^2+13y^2 (A033210) are congruent to 1, 9, 17, 25, 29,or 49 (mod 52). Cox lists some of these congruences on page 36 of his book. As mentioned by Cox, for these N, every term of the congruence has the form b^2 or N+b^2 for some integer b. In some cases, the congruences can be simplified. For instance, for N=18 (A106950), the congruence is 1, 19, 25, 43, 49, 67 (mod 72), which can be simplified to 1, 19 (mod 24).

			1, 2,
1, 2, 3,
1, 3, 7,
1, 5, 9, 13,
1, 5, 9,
1, 7,
1, 7, 9, 11, 15, 23, 25,
1, 9, 17, 25,
1, 13, 25,
1, 9, 11, 19,
1, 13, 25, 37,
1, 9, 13, 17, 25, 29, 49,
1, 19, 31, 49,
1, 9, 17, 25, 33, 41, 49, 57,
1, 19, 25, 43, 49, 67,
1, 25, 37,
1, 9, 15, 23, 25, 31, 47, 49, 71, 81,
1, 25, 49, 73,
...

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 3.

T. D. Noe, Rows n=1..65 of triangle, flattened
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

See the Binary Quadratic Forms and OEIS link for full list of primes generated by x^2+Ny^2, where N is a convenient number.

A139826 Squarefree idoneal numbers (A000926).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 13, 15, 21, 22, 30, 33, 37, 42, 57, 58, 70, 78, 85, 93, 102, 105, 130, 133, 165, 177, 190, 210, 253, 273, 330, 345, 357, 385, 462, 1365

Offset: 1

T. D. Noe, May 06 2008

A229462 Idoneal numbers (A000926) that are not pentagon exception numbers (A229461).

Original entry on oeis.org

7, 15, 28, 60, 112, 240

Offset: 1

Suggested by Eike Hertel, Hugo Pfoertner, Sep 24 2013

See A229461.

Idoneal numbers not of the form 7 or 15 times a square. - Hugo Pfoertner, Aug 13 2017

Eike Hertel, Reguläre Dreieckspflasterungen konvexer Polygone, Jenaer Schriften zur Mathematik und Informatik Math/Inf/01/13, 2013 (Preprint).
Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral Triangles, Discrete Comput Geom (2014) 51:753-759.
MathOverflow, Which natural numbers are a square minus a sum of two squares?. Answer by user "Lucia", 2014 Dec 5.

Cf. A000926 (idoneal numbers), A229461 (pentagon exception numbers).

A340132 Least prime numbers, in ascending order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes all the values of the sequence A000926 (idoneal numbers).

Original entry on oeis.org

1083289, 3818929, 6104641, 6868801, 7623529, 8465209, 9033649, 10105489, 11400481, 11597569, 11809561, 12338041, 12348961, 13154761, 13426009, 15861169, 16889161, 16922161, 18596449, 19684729, 20322481, 21067201, 21480001, 22684561, 23654569, 24531049

Offset: 1

Marco Frigerio, Dec 29 2020

nonn

First number in this sequence is equal to last number of sequence A338088.

The sequence is obtained using Lista(m), with m=246*10^5, see section PROG. It's possible to increase m to discover more terms of the sequence.

			1083289 =  315^2  + A000926(1)*992^2
        = 1033^2  + A000926(2)*90^2
        =  979^2  + A000926(3)*204^2
        = ...
        =  817^2  + A000926(65)*15^2.

Cf. A000926, A338088.

Idoneal()={return(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848]));}
isok(p,u)={my (i, s, n=matsize(u)[2], t=0);for(i=1, n, s=kronecker(-u[i],p); if(s==1, t++,break));if(t==n,t=0;for(i=1, n, s=qfbsolve(Qfb(1,0,u[i]),p); if(s==[], break,t++)));if(t==n,1,0)}
Primo(p, m)={my(u=Idoneal()); while(pr,v=concat(v,q),q=m)); return(v);}

A074402 Multiplicative closure of Euler's "numeri idonei" (A000926).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 37, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 57, 58, 60, 63, 64, 65, 66, 70, 72, 74, 75, 78, 80, 81, 84, 85, 88, 90, 91, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110

Offset: 1

Reinhard Zumkeller, Sep 25 2002

nonn

			22 is a term as A000926(17) = 22; 55 is not a term as 55 = 5*11 and neither 5 nor 11 are in A000926; 110 = 5*22 = A000926(5)*A000926(17), therefore 110 is a term.

A093668 Duplicate of A000926.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45

Offset: 1

dead

A340133 The sequence lists the least prime numbers, in ascending order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes all the values of the sequences A000926 (Idoneal numbers) and A003173 (Heegner numbers). See example.

Original entry on oeis.org

3230498881, 5086789009, 6956459689, 7260636769, 12387462649, 13125124321, 14049841129, 14247509329, 14310889849, 15871864849, 16573389361, 17502040609, 17768627809, 22042168201, 22621870441, 22957650769, 23018043409, 23819076121, 25228204849, 26585136601

Offset: 1

Marco Frigerio, Dec 29 2020

nonn

First number in this sequence is equal to least common number of sequences A340055 and A340132.

The sequence is obtained using Lista(m), with m=266*10^8, see section PROG. It's possible increase m to discover more terms of the sequence. It's also possible to extend the sequences A340055 and A340132 to check their common numbers.

			3230498881 = 2465^2+A000926(1)*56784^2
           = 56609^2+A000926(2)*3600^2
           = 35927^2+A000926(3)*25428^2
           = ...
           = 56791^2+A003173(9)*180^2
           = ...
           = 35743^2+A000926(65)*1028^2

Cf. A000926, A003173, A340055, A340132.

Union()={ my (v);v=(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848]));for(k=3, 41, d=4*k-1; if(isprime(d) && qfbclassno(-d)==1, v=concat(v, d)));return(v);}
isok(p,u)={my (i, s, n=matsize(u)[2], t=0);for(i=1, n, s=kronecker(-u[i],p); if(s==1, t++,break));if(t==n,t=0;for(i=1, n, s=qfbsolve(Qfb(1,0,u[i]),p); if(s==[], break,t++)));if(t==n,1,0)}
Primo(p, m)={my(u=Union()); while(pr,v=concat(v,q),q=m)); return(v);}

A139827 Primes of the form 2x^2 + 2xy + 17y^2.

Original entry on oeis.org

2, 17, 29, 41, 101, 149, 173, 197, 233, 281, 293, 461, 557, 569, 593, 677, 701, 761, 809, 821, 857, 941, 953, 1097, 1217, 1229, 1289, 1361, 1481, 1493, 1553, 1601, 1613, 1733, 1877, 1889, 1913, 1949, 1997, 2081, 2129, 2141, 2153, 2213, 2273, 2309, 2393, 2417

Offset: 1

T. D. Noe, May 02 2008, May 07 2008

Discriminant = -132.
Consider the quadratic form f(x,y) = ax^2 + bxy + cy^2. When the discriminant d=b^2-4ac is -4 times an idoneal number (A000926), there is exactly one class for each genus. As a result, the primes generated by f(x,y) are the same as the primes congruent to S (mod -d), where S is a set of numbers less than -d. The table on page 60 of Cox shows that there are exactly 331 quadratic forms having this property. The 217 sequences starting with this one complete the collection in the OEIS.
When a=1 and b=0, f(x,y) is a quadratic form whose congruences are discussed in A139642. Let N be an idoneal number. Then there are 2^r reduced quadratic forms whose discriminant is -4N, where r=1,2,3, or 4. By collecting the residuals p (mod 4N) for primes p generated by the i-th reduced quadratic form, we can empirically find a set Si. To show that the 2^r sets Si are complete, we only need to show that the union of the Si is equal to the set of numbers k such that the Jacobi symbol (-k/4N)=1.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, A139841-A139843 (d=-408), A139644, A139844-A139850 (d=-420), A139645, A139851-A139853 (d=-448), A139502, A139854-A139860 (d=-480), A139646, A139861-A139863 (d=-520), A139647, A139864-A139866 (d=-532), A139648, A139867-A139873 (d=-660), A139506, A139874-A139880 (d=-672), A139649, A139881-A139883 (d=-708), A139650, A139884-A139886 (d=-760), A139651, A139887-A139893 (d=-840), A139652, A139894-A139896 (d=-928), A139502, A139855, A139857, A139858, A139897-A139899, A139902 (d=-960).
Cf. also  A139653, A139904-A139906 (d=-1012), A139654, A139907-A139913 (d=-1092), A139655, A139914-A139920 (d=-1120), A139656, A139921-A139927 (d=-1248), A139657, A139928-A139934 (d=-1320), A139658, A139935-A139941 (d=-1380), A139659, A139942-A139948 (d=-1428), A139660, A139949-A139955 (d=-1540), A139661, A139956-A139962 (d=-1632), A139662, A139963-A139969 (d=-1848), A139663, A139970-A139976 (d=-2080), A139664, A139977-A139983 (d=-3040), A139665, A139984-A139998 (d=-3360), A139666, A139999-A140013 (d=-5280), A139667, A140014-A140028 (d=-5460), A139668, A140029-A140043 (d=-7392).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[ p: p in PrimesUpTo(2500) | p mod 132 in {2, 17, 29, 41, 65, 101}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[2, -2, 17, 2500] (* see A106856 *)
t = Table[{2, 17, 29, 41, 65, 101} + 132*n, {n, 0, 50}]; Select[Flatten[t], PrimeQ] (* T. D. Noe, Jun 21 2012 *)

v=[2, 17, 29, 41, 65, 101]; select(p->setsearch(v,p%132),primes(100)) \\ Charles R Greathouse IV, Jan 08 2013

The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).

A139643 Primes of the form x^2+Ny^2, with N=102.

Original entry on oeis.org

103, 127, 151, 223, 271, 409, 433, 457, 463, 577, 631, 727, 769, 919, 937, 967, 1033, 1039, 1063, 1087, 1249, 1279, 1327, 1447, 1471, 1543, 1657, 1753, 1759, 1777, 1783, 1801, 1879, 1951, 1993, 2089, 2143, 2161, 2287, 2311, 2473, 2503, 2551

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-408. N is an idoneal number (A000926), which means that the quadratic form's genus consists of a single class, which means that the primes of this form are identical to the primes that are congruent to c (mod 4N), where c is a set of numbers less than 4N. The sequence A139642 lists the set c for each idoneal number. That sequence also cross references the sequences for the quadratic forms with N equal to the first 36 idoneal numbers. The remaining quadratic forms are this sequence and the 28 listed in order below. Note that the sequences for N=120 and 240 are the same.

The primes are congruent to {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361} (mod 408).

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol 3, Chelsea, 1923.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139644, A139645, A139502, A139646, A139647, A139648, A139506, A139649, A139650, A139651, A139652, A139502, A139653-A139668.

[ p: p in PrimesUpTo(3000) | p mod 408 in {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361}]; // Vincenzo Librandi, Jul 28 2012

k:=102; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

C:= [1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361]:
select(isprime, [seq(seq(408*i+j,j=C),i=0..100)]); # Robert Israel, Jul 03 2016

nn=102; pMax=10000; Union[Reap[Do[p=x^2+nn*y^2; If[p<=pMax && PrimeQ[p], Sow[p]], {x,Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2,1]]] (* T. D. Noe, Aug 02 2009 *)
QuadPrimes2[1, 0, 102, 10000] (* see A106856 *)

A107008 Primes of the form x^2 + 24*y^2.

Original entry on oeis.org

73, 97, 193, 241, 313, 337, 409, 433, 457, 577, 601, 673, 769, 937, 1009, 1033, 1129, 1153, 1201, 1249, 1297, 1321, 1489, 1609, 1657, 1753, 1777, 1801, 1873, 1993, 2017, 2089, 2113, 2137, 2161, 2281, 2377, 2473, 2521, 2593, 2617, 2689, 2713

Offset: 1

T. D. Noe, May 09 2005

Presumably this is the same as primes congruent to 1 mod 24, so a(n) = 24*A111174(n) + 1. - N. J. A. Sloane, Jul 11 2008. Checked for all terms up to 2 million. - Vladimir Joseph Stephan Orlovsky, May 18 2011.
Discriminant = -96.
Also primes of the forms x^2 + 48*y^2 and x^2 + 72*y^2. See A140633. - T. D. Noe, May 19 2008
Primes of the quadratic form are a subset of the primes congruent to 1 (mod 24). [Proof. For 0 <= x, y <= 23, the only values mod 24 that x^2 + 24*y^2 can take are 0, 1, 4, 9, 12 or 16. All of these r except 1 have gcd(r, 24) > 1 so if x^2 + 24*y^2 is prime its remainder mod 24 must be 1.] - David A. Corneth, Jun 08 2020
More advanced mathematics seems to be needed to determine whether this sequence lists all primes congruent to 1 (mod 24). Note the significance of 24 being a convenient number, as described in A000926. See also Sloane et al., Binary Quadratic Forms and OEIS, which explains how the table in A139642 may be used for this determination. - Peter Munn, Jun 21 2020
Primes == 1 (mod 2^3*3) are the intersection of the primes == 1 (mod 2^3) in A007519 and the primes == 1 (mod 3) in A002476, by the Chinese remainder theorem. - R. J. Mathar, Jun 11 2020

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 143 terms from N. J. A. Sloane]
P. L. Clark, J. Hicks, H. Parshall, K. Thompson, GONI: primes represented by binary quadratic forms, INTEGERS 13 (2013) #A37
D. A. Cox, Primes of the form x^2 + n*y^2, A Wiley-Interscience publication, 1989
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. Voight, Quadratic forms that represent almost the same primes, Math. Comp. 76 (2007) 1589-1617

Subset of A033199 (2y here = y there).
Is this the same as A141375?
Cf. A000926, A002476, A007519, A111174, A139642.
See also the cross-references in A140633.

QuadPrimes[1, 0, 24, 10000] (* see A106856 *)

is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ David A. Corneth, Jun 21 2020

Recomputed b-file, deleted incorrect Mma program. - N. J. A. Sloane, Jun 08 2014

cp's OEIS Frontend

A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

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A139826 Squarefree idoneal numbers (A000926).

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A229462 Idoneal numbers (A000926) that are not pentagon exception numbers (A229461).

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A340132 Least prime numbers, in ascending order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes all the values of the sequence A000926 (idoneal numbers).

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A074402 Multiplicative closure of Euler's "numeri idonei" (A000926).

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A093668 Duplicate of A000926.

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A139827 Primes of the form 2x^2 + 2xy + 17y^2.

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Magma

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A139643 Primes of the form x^2+Ny^2, with N=102.

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Magma

Magma

Maple

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A107008 Primes of the form x^2 + 24*y^2.

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