A092572 -id:A092572 - cp's OEIS frontend

A155560 Intersection of A000404 and A092572: N = a^2 + b^2 = c^2 + 3d^2 with a,b,c,d>0.

13, 37, 52, 61, 73, 97, 100, 109, 117, 148, 157, 169, 181, 193, 208, 229, 241, 244, 277, 292, 313, 325, 333, 337, 349, 373, 388, 397, 400, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 628, 637, 657, 661, 673, 676, 709, 724, 733, 757, 769

Offset: 1

M. F. Hasler, Jan 24 2009

nonn

Nonsquare terms of A155563. - Joerg Arndt, Jan 11 2015

			a(1)=13 is the least number that can be written as A+B and C+3D where A,B,C,D are positive squares (namely 13 = 2^2 + 3^2 = 1^2 + 3*2^2).
a(2)=37 is the second smallest number which figures in A000404 and in A092572 as well.

isA155560(n /* omit optional 2nd arg for the present sequence */, c=[3,1]) = { for(i=1,#c,for(b=1,sqrtint((n-1)\c[i]),issquare(n-c[i]*b^2)&next(2));return);1}
for( n=1,10^3, isA155560(n) & print1(n","))

is(n)=!issquare(n) && #bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n);
select(n->is(n), vector(1500,j,j)) \\ Joerg Arndt, Jan 11 2015

A155712 Intersection of A092572 and A155716: N = a^2 + 3b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

7, 28, 31, 49, 63, 73, 79, 97, 100, 103, 112, 124, 127, 151, 175, 193, 196, 199, 217, 223, 241, 252, 271, 279, 292, 313, 316, 337, 343, 367, 388, 400, 409, 412, 433, 439, 441, 448, 457, 463, 484, 487, 496, 508, 511, 553, 567, 577, 601, 604, 607, 631, 657, 673

Offset: 1

M. F. Hasler, Jan 25 2009

From Robert Israel, Jan 19 2025: (Start)
If k is a term, then so is j^2 * k for all positive integers j.
The primes in this sequence appear to be A033199.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000404, A033199, A092572, A097268, A154777, A154778, A155716, A155717, A155560-A155578.

N:= 1000: # for terms <= N
A:= {seq(seq(a^2 + 3*b^2, b=1 .. floor(sqrt((N-a^2)/3))),a=1..floor(sqrt(N)))}
   intersect {seq(seq(c^2 + 6*d^2, d = 1 .. floor(sqrt((N-c^2)/6))),c=1..floor(sqrt(N)))}:
sort(convert(A,list)); # Robert Israel, Jan 19 2025

isA155712(n,/* optional 2nd arg allows to get other sequences */c=[6,3]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) && next(2)); return);1}
for( n=1,999, isA155712(n) && print1(n",")) \\ Update to modern PARI syntax (& -> &&) by M. F. Hasler, Jan 18 2025

A155574 Intersection of A154777 and A092572: N = a^2 + 2b^2 = c^2 + 3d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

12, 19, 36, 43, 48, 57, 67, 73, 76, 97, 108, 129, 139, 144, 147, 163, 171, 172, 192, 193, 201, 211, 219, 228, 241, 268, 283, 291, 292, 300, 304, 307, 313, 324, 331, 337, 361, 379, 387, 388, 409, 417, 432, 433, 441, 457, 475, 484, 489, 499, 507, 513, 516, 523

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155564 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155574(n,/* optional 2nd arg allows us to get other sequences */c=[3,2]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155574(n) & print1(n","))

A155710 Intersection of A092572 and A154778: N = a^2 + 3b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

21, 36, 49, 61, 84, 109, 129, 144, 181, 189, 196, 201, 229, 241, 244, 301, 309, 324, 336, 349, 381, 409, 421, 436, 441, 469, 489, 516, 525, 541, 549, 576, 601, 661, 669, 709, 721, 724, 756, 769, 784, 804, 829, 849, 889, 900, 916, 921, 964, 976, 981, 1009, 1021

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155570 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

isA155710(n,/* use optional 2nd arg to get other analogous sequences */c=[5,3]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1111, isA155710(n) & print1(n","))

A380295 Numbers that can be written as a^2 + 3b^2 for some a, b in A155716 and also as c^2 + 6d^2 for some c, d in A092572.

Original entry on oeis.org

1552, 1975, 4753, 5047, 5425, 7825, 8167, 9175, 10096, 11025, 11536, 12007, 16528, 16807, 16993, 18823, 19600, 23863, 24832, 25633, 25767, 26983, 27223, 29200, 30919, 31600, 31927, 32791, 33175, 35329, 35623, 41953, 43063, 43687, 51943, 54775, 57303, 59575, 60016, 61783, 63175, 71575, 72103

Offset: 1

Robert Israel, Jan 19 2025

nonn

If k is a term, then so is j^4 * k for all positive integers j.

			a(5) = 5425 is a term because 5425 = 73^2 + 6 * 4^2 = 25^2 + 3 * 40^2 with 73 = 5^2 + 3 * 4^2, 4 = 1^2 + 3 * 1^2, 25 = 1^2 + 6 * 2^2 and 40 = 4^2 + 6 * 2^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A092572, A155712, A155716.

N:= 500: # for terms <= N^2
A:= {seq(seq(a^2 + 3*b^2,a=1..floor(sqrt(N-3*b^2))),b=1..floor(sqrt(N/3)))}:
B:= {seq(seq(a^2 + 6*b^2,a=1..floor(sqrt(N-6*b^2))),b=1..floor(sqrt(N/6)))}:
C:= select(`<=`,{seq(seq(a^2+6*b^2,a=A),b=A)},N^2):
E:= select(`<=`,{seq(seq(a^2+3*b^2,a=B),b=B)},N^2):
sort(convert(C intersect E,list));

A155571 Intersection of A000404, A092572 and A154778: N = a^2 + b^2 = c^2 + 3d^2 = e^2 + 5f^2 for some positive integers a,b,c,d,e,f.

Original entry on oeis.org

61, 109, 181, 229, 241, 244, 349, 409, 421, 436, 541, 549, 601, 661, 709, 724, 769, 829, 900, 916, 964, 976, 981, 1009, 1021, 1069, 1129, 1201, 1225, 1249, 1321, 1381, 1396, 1429, 1489, 1521, 1525, 1549, 1609, 1621, 1629, 1636, 1669, 1684, 1741, 1744, 1789

Offset: 1

M. F. Hasler, Jan 25 2009

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155571(n,/* optional 2nd arg allows us to get other sequences */c=[5,3,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1999, isA155571(n) & print1(n","))

A155573 Intersection of A000404, A154777 and A092572: N = a^2 + b^2 = c^2 + 2d^2 = e^2 + 3f^2 for some positive integers a,b,c,d,e,f.

Original entry on oeis.org

73, 97, 193, 241, 292, 313, 337, 388, 409, 433, 457, 577, 601, 657, 673, 769, 772, 873, 900, 937, 964, 1009, 1033, 1129, 1153, 1156, 1168, 1201, 1249, 1252, 1297, 1321, 1348, 1489, 1521, 1552, 1609, 1636, 1657, 1732, 1737, 1753, 1777, 1801, 1825, 1828

Offset: 1

M. F. Hasler, Jan 25 2009

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155573(n,/* optional 2nd arg allows us to get other sequences */c=[3,2,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1999, isA155573(n) & print1(n","))

A003136 Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192

Offset: 1

N. J. A. Sloane

Equally, numbers of the form x^2 - xy + y^2. - Ray Chandler, Jan 27 2009
Also, numbers of the form X^2+3Y^2 (X=y+x/2, Y=x/2), cf. A092572. - Zak Seidov, Jan 20 2009
Theorem (Schering, Delone, Watson): The only positive definite binary quadratic forms that represent the same numbers are x^2+xy+y^2 and x^2+3y^2 (up to scaling). - N. J. A. Sloane, Jun 22 2014
Equivalently, numbers n such that the coefficient of x^n in Theta3(x)*Theta3(x^3) is nonzero. - Joerg Arndt, Jan 16 2011
Equivalently, numbers n such that the coefficient of x^n in a(x) (resp. b(x)) is nonzero where a(), b() are cubic AGM functions. - Michael Somos, Jan 16 2011
Relative areas of equilateral triangles whose vertices are on a triangular lattice. - Anton Sherwood (bronto(AT)pobox.com), Apr 05 2001
2 appended to a(n) (for positive n) corresponds to capsomere count in viral architectural structures (cf. A071336). - Lekraj Beedassy, Apr 14 2006
The triangle in A132111 gives the enumeration: n^2 + k*n + k^2, 0 <= k <= n.
The number of coat proteins at each corner of a triangular face of a virus shell. - Parthasarathy Nambi, Sep 04 2007
Numbers of the form (x^2 + y^2 + (x + y)^2)/2. If we let z = - x - y, then all the solutions to x^2 + y^2 + z^2 = k with x + y + z = 0 are k = 2a(n) for any n. - Jon Perry, Dec 16 2012
Sequence of divisors of the hexagonal lattice, except zero (where it is said that an integer n divides a lattice if there exists a sublattice of index n; example: 3 divides the hexagonal lattice). - Jean-Christophe Hervé, May 01 2013
Numbers of the form - (x*y + y*z + x*z) with x + y + z = 0. Numbers of the form x^2 + y^2 + z^2 - (x*y + y*z + x*z) = (x - y)*(x - z) + (y - x) * (y - z) + (z - x) * (z - y). - Michael Somos, Jun 26 2013
Equivalently, the existence spectrum of affine Mendelsohn triple systems, cf. A248107. - David Stanovsky, Nov 25 2014
Lame's solutions to the Helmholtz equation with Dirichlet boundary conditions on the unit-edged equilateral triangle have eigenvalues of the form: (x^2+x*y+y^2)*(4*Pi/3)^2. The actual set, starting at 1 and counting degeneracies, is given by A060428, e.g., the first degeneracy is 49 where (x,y)=(0,7) and (3,5). - Robert Stephen Jones, Oct 01 2015
Curvatures of spheres in one bowl of integers, the Loeschian spheres. Mod 12, numbers equal to 0, 1, 3, 4, 7, 9. - Ed Pegg Jr, Jan 10 2017
Norms of Eisenstein integers Z[omega] or k(rho). - Juris Evertovskis, Dec 07 2017
Named after the German economist August Lösch (1906-1945). - Amiram Eldar, Jun 10 2021
Starting from the second element, these and only these numbers of congruent equilateral triangles can be used to cover a regular tetrahedron without overlaps or gaps. - Alexander M. Domashenko, Feb 01 2025
This sequence is closed under multiplication: (x; y)*(u; v) = (x*v - y*u; x*u + y*(u + v)) for x*v - y*u >= 0 , (x; y)*(u; v) = (y*u - x*v; x*u + v*(x + y)) for x*v - y*u < 0. - Alexander M. Domashenko, Feb 03 2025

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 111.
Ivars Peterson, The Jungles of Randomness: A Mathematical Safari, John Wiley and Sons, (1998) pp. 53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Shreeya Behera, Variants of the chromatic number of the plane, Ph. D. dissertation, Ohio State Univ., 2024. See p. 88.
Mira Bernstein, N. J. A. Sloane and Paul E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math., Vol. 170, No. 1-3 (1997), pp. 29-39; (Abstract, pdf, ps).
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
John H. Conway, E. M. Rains, and N. J. A. Sloane, On the existence of similar sublattices, Canadian Journal of Mathematics, Vol. 51, No. 6 (1999), pp. 1300-1306; (Abstract, pdf, ps).
B. N. Delone, Geometry of positive quadratic forms. Part II, Uspekhi Mat. Nauk, Vol. 4 (1938), pp. 102-164.
Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal and David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], (19-November-2014).
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
A. Itin, New geometric receipts for design of photonic crystals and metamaterials: optimal toric packings, arXiv:2410.17424 [physics.app-ph], 2024. See p. 4.
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]
Yoshiyuki Kitaoka, On the relation between the positive-definite quadratic forms with the same representation numbers, Proc. Jap. Acad., Vol. 47 (1971), pp. 439-441.
August Lösch, Economics of Location, New Haven and London: Yale University Press, 1954. See pp. 117f.
Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.
John U. Marshall, The Löschian numbers as a problem in number theory, Geographical Analysis, Vol. 7, No. 4 (1975), pp. 421-426; Annotated scanned copy.
John U. Marshall, The construction of the Löschian landscape, Geographical Analysis, Vol. 9, No. 1 (Jan. 1977), pp. 1-13; Annotated scanned copy.
John U. Marshall, Christallerian networks in the Löschian economic landscape, Professional Geographer, Vol. 29, No. 2 (1977), pp. 153-159; Annotated scanned copy.
John U. Marshall, Letter to N. J. A. Sloane, 1990.
Umesh P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
Ed Pegg, Jr., Loeschian Spheres, 2015.
Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.
Katherine A. Ritchey, Computational Topology for Configuration Spaces of Disks in a Torus, Ph. D. Dissertation, The Ohio State University (2019).
Jacob Rus, Flowsnake Earth, Bridges 2017 Conference Proceedings, pp. 237-244.
M. Schering, Théorèmes relatifs aux formes quadratiques qui représentent les mêmes nombres, J. Math, pures el appl., 2 serie 4 (1859).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
James Smith, Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling, arXiv:2403.01911 [math.MG], 2024. See p. 18.
V. N. Timofeev, On positive quadratic forms, representing the same numbers, Uspekhi Mat. Nauk, Vol. 18 (1963), pp. 191-193.
G. L. Watson, Determination of a binary quadratic form by its values at integer points, Mathematika, Vol. 26, No. 1 (1979), pp. 72-75. MR0557128 (81e:10019).
G. L. Watson, Acknowledgement: Determination of a binary quadratic form by its values at integer points, Mathematika, Vol. 27, No. 2 (1980),p. 188. MR0610704 (82d:10037)
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for "core" sequences

Subsequence of A032766, A198772, A118886, A198773, A198774, A198775, A202822, and A260682.
See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.
See A088534 for the number of representations.
Cf. A034020 (complement), A007645 (primes); partitions: A198726, A198727.
Cf. A004611, A034017, A045897, A060428.

import Data.Set (singleton, union, fromList, deleteFindMin)
a003136 n = a003136_list !! (n-1)
a003136_list = f 0 $ singleton 0 where
f x s | m < x ^ 2 = m : f x s'
| otherwise = m : f x'
(union s' $ fromList $ map (\y -> x'^2+(x'+y)*y) [0..x'])
where x' = x + 1
(m,s') = deleteFindMin s
-- Reinhard Zumkeller, Oct 30 2011

function isA003136(n)
    n % 3 == 2 && return false
    n in [0, 1, 3] && return true
    M = Int(round(2*sqrt(n/3)))
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && return true
    end
    return false
end
A003136list(upto) = [n for n in 0:upto if isA003136(n)]
A003136list(192) |> println # Peter Luschny, Mar 17 2018

[n: n in [0..192] | NormEquation(3, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

readlib(ifactors): for n from 2 to 200 do m := ifactors(n)[2]: flag := 1: for i from 1 to nops(m) do if m[i,1] mod 3 = 2 and m[i,2] mod 2 = 1 then flag := 0; break fi: od: if flag=1 then printf(`%d,`,n) fi: od: # James Sellers, Dec 07 2000

ok[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]]; Select[Range[0, 192], ok] (* Jean-François Alcover, Apr 18 2011 *)
nn = 14; Select[Union[Flatten[Table[x^2 + x*y + y^2, {x, 0, nn}, {y, 0, x}]]], # <= nn^2 &] (* T. D. Noe, Apr 18 2011 *)
QP = QPochhammer; s = QP[q]^3 / QP[q^3]/3 + O[q]^200; Position[ CoefficientList[s, q], n_ /; n != 0] - 1 // Flatten (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

isA003136(n)=local(fac,flag);if(n==0,1,fac=factor(n);flag=1;for(i=1,matsize(fac)[1],if(Mod(fac[i,1],3)==2 && Mod(fac[i,2],2)==1,flag=0));flag)

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

x='x+O('x^200); p=eta(x)^3/eta(x^3); for(n=0, 199, if(polcoeff(p, n) != 0, print1(n, ", "))) \\ Altug Alkan, Nov 08 2015

list(lim)=my(v=List(),y,t); for(x=0,sqrtint(lim\3), my(y=x,t); while((t=x^2+x*y+y^2)<=lim, listput(v,t); y++)); Set(v) \\ Charles R Greathouse IV, Feb 05 2017

is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3) \\ Hugo Pfoertner, Aug 04 2023

from itertools import count, islice
from sympy import factorint
def A003136_gen(): return (n for n in count(0) if all(e % 2 == 0 for p,e in factorint(n).items() if p % 3 == 2))
A003136_list = list(islice(A003136_gen(),30)) # Chai Wah Wu, Jan 20 2022

Either n=0 or else in the prime factorization of n all primes of the form 3a+2 must occur to even powers only (there is no restriction of primes congruent to 0 or 1 mod 3).
If n is in the sequence, then n^k is in the sequence (but the converse is not true). n is in the sequence iff n^(2k+1) is in the sequence. - Ray Chandler, Feb 03 2009
A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
The sequence is multiplicative in the sense that if m and n are in the sequence, so is m*n. - Jon Perry, Dec 18 2012
Comments from Richard C. Schroeppel, Jul 20 2016: (Start)
The set is also closed under restricted division: If M and N are members, and M divides N, then N/M is a member.
If N == 2 (mod 3), N is not in the sequence.
The density of members (relative to the integers>0) gradually falls to 0. The density goes as O(1/sqrt(log N)). This implies that, if N is a member, the average expected number of representations of N is O(sqrt(log N)).
Representations usually come in sets of 6: (K,L), (K+L,-K), (K+L,-L) and their negatives. (End)
Since Q(zeta), where zeta is a primitive 3rd root of unity has class number 1, the situation as to whether an integer is of the form x^2 + xy + y^2 is similar to the situation with x^2 + y^2: n is of that form if and only if every prime p dividing n which is = 5 mod 6 divides it to an even power. The density of 1/sqrt(x) that Rich mentioned is an old result due to Landau. - Victor S. Miller, Jul 20 2016
From Juris Evertovskis, Dec 07 2017; Jan 01 2020: (Start)
In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2+xy+y^2=n is 6*Product_{p_i in S_1} (e_i + 1) if all e_j are even and 0 otherwise.
For all Löschian numbers there are nonnegative X,Y such that X^2+XY+Y^2=n. For x,y such that x^2+xy+y^2=n take X=min(|x|,|y|), Y=|x+y| if xy<0 and X=|x|, Y=|y| otherwise. (End)

A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

Original entry on oeis.org

5, 8, 13, 17, 20, 25, 29, 32, 37, 40, 41, 45, 52, 53, 61, 65, 68, 72, 73, 80, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 125, 128, 136, 137, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 197, 200, 205, 208, 212, 221, 225, 229, 232, 233

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and A024352 (difference of squares).

Also: Numbers of the form x^2+4y^2, where x and y are positive integers. Cf. A154777, A092572, A154778 for analogous sequences. - M. F. Hasler, Jan 24 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097269, A097270, A097271.

isA097268(n) = forstep( b=2,sqrtint(n-1),2, issquare(n-b^2) && return(1)) \\ M. F. Hasler, Jan 24 2009

A155716 Numbers of the form N = a^2 + 6b^2 for some positive integers a,b.

Original entry on oeis.org

7, 10, 15, 22, 25, 28, 31, 33, 40, 42, 49, 55, 58, 60, 63, 70, 73, 79, 87, 88, 90, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 145, 150, 151, 154, 159, 160, 166, 168, 175, 177, 186, 193, 196, 198, 199, 202, 214, 217, 220, 223, 225, 231, 232, 240

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A002481 (which allows for a and b to be zero).

Primes are in A033199. - Bernard Schott, Sep 20 2019

David A. Corneth, Table of n, a(n) for n = 1..17743 (terms <= 10^5)

Cf. A002481, A033199.

Cf. A000404, A154777, A092572, A097268, A154778, A155707-A155717, A155560-A155578.

With[{upto=240},Select[Union[#[[1]]^2+6#[[2]]^2&/@Tuples[ Range[Sqrt[ upto]], 2]],#<=upto&]] (* Harvey P. Dale, Aug 05 2016 *)

isA155716(n,/* optional 2nd arg allows us to get other sequences */c=6) = { for(b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,999, isA155716(n) & print1(n","))

upto(n) = my(res=List()); for(i=1,sqrtint(n),for(j=1, sqrtint((n - i^2) \ 6), listput(res, i^2 + 6*j^2))); listsort(res,1); res \\ David A. Corneth, Sep 18 2019

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A155560 Intersection of A000404 and A092572: N = a^2 + b^2 = c^2 + 3d^2 with a,b,c,d>0.

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A155712 Intersection of A092572 and A155716: N = a^2 + 3b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

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A155574 Intersection of A154777 and A092572: N = a^2 + 2b^2 = c^2 + 3d^2 for some positive integers a,b,c,d.

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A155710 Intersection of A092572 and A154778: N = a^2 + 3b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

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A380295 Numbers that can be written as a^2 + 3*b^2 for some a, b in A155716 and also as c^2 + 6*d^2 for some c, d in A092572.

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A155571 Intersection of A000404, A092572 and A154778: N = a^2 + b^2 = c^2 + 3d^2 = e^2 + 5f^2 for some positive integers a,b,c,d,e,f.

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A155573 Intersection of A000404, A154777 and A092572: N = a^2 + b^2 = c^2 + 2d^2 = e^2 + 3f^2 for some positive integers a,b,c,d,e,f.

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A003136 Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.

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A380295 Numbers that can be written as a^2 + 3b^2 for some a, b in A155716 and also as c^2 + 6d^2 for some c, d in A092572.