A084916 -id:A084916 - cp's OEIS frontend

A232532 Positive numbers not of the form x^2 - 3*y^2; complement of A084916.

2, 3, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 23, 26, 27, 28, 29, 30, 31, 32, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 50, 51, 53, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 95, 98, 99

Offset: 1

V. Raman, Nov 25 2013

nonn

Previous name was: Numbers j such that the equation a^2 + 3*j*b^2 = 3*c^2 + j*d^2 has no solutions in positive integers for a, b, c, d.

If j is in the sequence, so is j*k^2 for any positive integer k. - Charles R Greathouse IV, Dec 13 2013

V. Raman, Proof for individual terms

Cf. A084916 (complement).

Cf. A232531, A232681, A232682.

for(n=1,99, if ([]==qfbsolve(Qfb(1,0,-3),n,2),print1(n,", "))); \\ Joerg Arndt, Jun 05 2022

Six more terms from V. Raman, Dec 13 2013

Edited by and better name from Jon E. Schoenfield and Joerg Arndt, Jun 05 2022

A068228 Primes congruent to 1 (mod 12).

Original entry on oeis.org

13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

This has several equivalent definitions (cf. the Tunnell link)
Also primes of the form x^2 + 9y^2 (discriminant -36). - T. D. Noe, May 07 2005 [corrected by Klaus Purath, Jan 18 2023]
Also primes of the form x^2 - 12y^2 (discriminant 48). Cf. A140633. - T. D. Noe, May 19 2008 [corrected by Klaus Purath, Jan 18 2023]
Also primes of the form x^2 + 4*x*y + y^2.
Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916).
Also primes of the form x^2 + 6*x*y - 3*y^2.
Also primes of the form 4*x^2 + 8*x*y + y^2.
Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - Tito Piezas III, Dec 28 2008
Sequence lists generalized cuban primes (A007645) that are the sum of 2 nonzero squares. - Altug Alkan, Nov 25 2015
Yasutoshi Kohmoto observes that prevprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the previous prime must be at a gap of 4 or 8 or 12 ..., but a gap of 4 is impossible because 12k + 1 - 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the previous prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs. 65% as the above simple explanation suggests, but considering primes up to 10^8 yields a ratio of about 41% vs. 59%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017
Also primes of the form x^2 - 27*y^2. - Klaus Purath, Jan 18 2023

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy, with permission]
Michael Penn, an example right from my number theory class., YouTube video, 2021.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
J. Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732.
Subsequence of A084916.
Subsequence of A007645.
Also primes in A084916, A020672.
Cf. A141123 (d=12), A141111, A141112 (d=65), A141187 (d=48) A038872 (d=5), A038873 (d=8), A038883 (d=13), A038889 (d=17).
For a 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(1400) | p mod 12 in {1}]; // Vincenzo Librandi, Jul 14 2012
For other programs see the "Binary Quadratic Forms and OEIS" link.

select(isprime, [seq(i,i=1..10000, 12)]); # Robert Israel, Nov 27 2015

Select[Prime/@Range[250], Mod[ #, 12]==1&]
Select[Range[13, 10^4, 12], PrimeQ] (* Zak Seidov, Mar 21 2011 *)

for(i=1,250, if(prime(i)%12==1, print(prime(i))))

forstep(p=13,10^4,12,isprime(p)&print(p)); \\ Zak Seidov, Mar 21 2011

Edited by Dean Hickerson, Feb 27 2002

Entry revised by N. J. A. Sloane, Oct 18 2014 (Edited, merged with A141122, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008).

A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

Original entry on oeis.org

1, 4, 5, 9, 11, 16, 19, 20, 25, 29, 31, 36, 41, 44, 45, 49, 55, 59, 61, 64, 71, 76, 79, 80, 81, 89, 95, 99, 100, 101, 109, 116, 121, 124, 125, 131, 139, 144, 145, 149, 151, 155, 164, 169, 171, 176, 179, 180, 181, 191, 196, 199, 205, 209, 211, 220, 225, 229, 236

Offset: 1

N. J. A. Sloane

5x^2 - y^2 has discriminant 20, x^2 + xy - y^2 has discriminant 5. - N. J. A. Sloane, May 30 2014
Representable as x^2 + 3xy + y^2 with 0 <= x <= y. - Benoit Cloitre, Nov 16 2003
Numbers k such that x^2 - 3xy + y^2 + k = 0 has integer solutions. - Colin Barker, Feb 04 2014
Numbers k such that x^2 - 7xy + y^2 + 9k = 0 has integer solutions. - Colin Barker, Feb 10 2014
Also positive numbers of the form x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

M. Baake, "Solution of coincidence problem ...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Robert Israel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 [math.MG], 2006.
M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Norbert Hungerbühler and Maciej Smela, Geometric approach to the Diophantine equation x^2 + x*y - y^2 = m, hal-04835410, 2024. See p. 18.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), this sequence, A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), A316621(k=9).
See A035187 for number of representations.
Primes in this sequence: A038872, also A141158.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
See also the related sequence A263849 based on a theorem of Maass.

select(t -> nops([isolve(5*x^2-y^2=t)])>0, [$1..1000]); # Robert Israel, Jun 12 2014

ok[n_] := Resolve[Exists[{x, y}, Element[x|y, Integers], n == 5*x^2-y^2]]; Select[Range[236], ok]
(* or, for a large number of terms: *)
max = 60755 (* max=60755 yields 10000 terms *); A031363 = {}; xm = 1;
While[T = A031363; A031363 = Table[5*x^2 - y^2, {x, 1, xm}, {y, 0, Floor[ x*Sqrt[5]]}] // Flatten // Union // Select[#, # <= max&]&; A031363 != T, xm = 2*xm]; A031363  (* Jean-François Alcover, Mar 21 2011, updated Mar 17 2018 *)

select(x -> x, direuler(p=2,101,1/(1-(kronecker(5,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020, after hints by Colin Barker, Jun 18 2014, and Michel Marcus

is(n)=#bnfisintnorm(bnfinit(z^2-z-1),n) \\ Ralf Stephan, Oct 18 2013

seq(M,k=3) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
seq(236) \\ Gheorghe Coserea, Jul 29 2018

from itertools import count, islice
from sympy import factorint
def A031363_gen(): # generator of terms
    return filter(lambda n:all(not((1 < p % 5 < 4) and e & 1) for p, e in factorint(n).items()),count(1))
A031363_list = list(islice(A031363_gen(),30)) # Chai Wah Wu, Jun 28 2022

Consists exactly of numbers in which primes == 2 or 3 mod 5 occur with even exponents.

Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = 5.

More terms from Erich Friedman

b-file corrected and extended by Robert Israel, Jun 12 2014

A243655 Positive numbers that are primitively represented by the indefinite quadratic form x^2 - 3y^2 of discriminant 12.

Original entry on oeis.org

1, 6, 13, 22, 33, 37, 46, 61, 69, 73, 78, 94, 97, 109, 118, 121, 141, 142, 157, 166, 169, 177, 181, 193, 213, 214, 222, 229, 241, 249, 253, 262, 277, 286, 313, 321, 334, 337, 349, 358, 366, 373, 382, 393, 397, 409, 421, 429, 433, 438, 454, 457, 478, 481

Offset: 1

N. J. A. Sloane, Jun 11 2014

nonn

x^2+2xy-2y^2 is an equivalent form.

Will Jagy, C++ program Conway_Positive_All.cc to find all positive numbers represented by an indefinite binary quadratic form
Will Jagy, Sample output from Conway_Positive_All.cc
Will Jagy, C++ program Conway_Positive_Primitive.cc to find positive numbers primitively represented by an indefinite binary quadratic form
Will Jagy, Sample output from Conway_Positive_Prim.cc
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A084916 (all numbers represented), A068228.

Reap[For[n = 1, n < 500, n++, r = Reduce[x^2 - 3 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

Original entry on oeis.org

13, 25, 37, 52, 61, 73, 97, 100, 109, 117, 148, 157, 169, 181, 193, 208, 225, 229, 241, 244, 277, 289, 292, 313, 325, 333, 337, 349, 373, 388, 397, 400, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 625, 628, 637, 657, 661, 673, 676, 709, 724, 733

Offset: 1

Altug Alkan, Nov 22 2015

nonn

n is in the sequence iff 4*n is.
If a(n) is a prime number, a(n) mod 12 = 1.
Prime terms of sequence are listed in A068228 that lists generalized cuban primes (A007645) which are the sum of 2 nonzero squares.
Also positive numbers of the form x^2 - 3*y^2 (A084916) that are the sum of 2 nonzero squares. - Frank M Jackson, Oct 13 2019

			a(1) = 13 because 13 = 3^2 + 3*1 + 1^2 = 3^2 + 2^2.
a(2) = 25 because 25 = 5^2 + 5*0 + 0^2 = 4^2 + 3^2.
a(3) = 37 because 37 = 4^2 + 4*3 + 3^2 = 6^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000404, A003136, A068228, A084916.

Select[Range@750, Length[PowersRepresentations[#, 2, 2] /. {0, }->Nothing]>0 && Reduce[#==x^2+x*y+y^2, {x, y}, Integers]=!=False &] (* _Frank M Jackson, Oct 13 2019 *)

isok(n) = { for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=1, 1e3, if( is(n) && isok(n), print1(n, ", ")))

A316621 Numbers of the form x^2 + 9xy + y^2, 0 <= x <= y.

Original entry on oeis.org

0, 1, 4, 9, 11, 16, 23, 25, 36, 37, 44, 49, 53, 64, 67, 71, 81, 91, 92, 99, 100, 113, 119, 121, 133, 137, 144, 148, 163, 169, 176, 179, 191, 196, 207, 212, 221, 225, 247, 253, 256, 268, 275, 284, 287, 289, 317, 323, 324, 331, 333, 361, 364, 368, 379, 389, 396, 400, 401, 407, 421, 427, 441, 443, 449

Offset: 1

Gheorghe Coserea, Jul 29 2018

nonn

Discriminant 77.

In general, for k>=0 the positive part of the set S = {x^2 - k*x*y + y^2: x,y in Z} is given by the numbers of the form x^2 + k*x*y + y^2 with 0 <= x <= y natural numbers.

Gheorghe Coserea, Table of n, a(n) for n = 1..100001

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), A031363(k=3), A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), this sequence.

seq(M,k=9) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
concat(0, seq(449))

cp's OEIS Frontend

A232532 Positive numbers not of the form x^2 - 3*y^2; complement of A084916.

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A068228 Primes congruent to 1 (mod 12).

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A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

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A243655 Positive numbers that are primitively represented by the indefinite quadratic form x^2 - 3y^2 of discriminant 12.

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Mathematica

A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

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A316621 Numbers of the form x^2 + 9*x*y + y^2, 0 <= x <= y.

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A316621 Numbers of the form x^2 + 9xy + y^2, 0 <= x <= y.