A202822 -id:A202822 - cp's OEIS frontend

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

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)

A260682 Löschian numbers (A003136) of the form 6*k+1.

Original entry on oeis.org

1, 7, 13, 19, 25, 31, 37, 43, 49, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 169, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 361, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499

Offset: 1

Joerg Arndt, Nov 15 2015

nonn

Odd terms of A202822, which lists Löschian numbers of the form 3*k+1. - Altug Alkan, Nov 15 2015

Reinhard Zumkeller, 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.

Cf. A003136, A016921, A202822.

a260682 n = a260682_list !! (n-1)
a260682_list = filter ((== 1) . flip mod 6) a003136_list
-- Reinhard Zumkeller, Nov 16 2015

nn = 25; Select[Union[Flatten[Table[x^2 + x*y + y^2, {x, 0, nn}, {y, 0, x}]]], Mod[#, 6] == 1 && # <= nn^2&] (* Jean-François Alcover, Jul 21 2018, after T. D. Noe *)

is(n)=(n%6==1)&&#bnfisintnorm(bnfinit(z^2+z+1), n);
select(n->is(n), vector(500,j,j))

x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%6==1, print1(n, ", "))) \\ Altug Alkan, Nov 15 2015

isok(n) = if( n<1 || (n%3 == 0), 0, 0 != sumdiv( n, d, kronecker( -3, d))) && n%2==1;
for(n=0, 500, if(isok(n), print1(n", "))) \\ Altug Alkan, Nov 15 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, if(t%6==1, listput(v, t)); y++)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

A267137 Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 20, 21, 22, 24, 25, 26, 30, 32, 33, 34, 36, 37, 40, 41, 42, 44, 46, 49, 50, 52, 54, 56, 57, 58, 60, 64, 65, 66, 69, 70, 72, 74, 76, 80, 81, 82, 85, 86, 89, 90, 92, 94, 96, 97, 100, 101, 102, 104, 105, 108, 110, 112, 114, 116

Offset: 1

Altug Alkan, Jan 10 2016

nonn

Inspired by relation between A003136 and A202822. See comment section of A202822.
Prime terms of this sequence are 2, 5, 17, 37, 41, 89, 97, 101, 137, 149, ...
Perfect power terms of this sequence are 1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 144, 169, ...
Obviously, A000290, A002378 and A045944 are subsequences.
The complement of this sequence is A322430. - Kemoneilwe Thabo Moseki, Dec 12 2019

			1 is a term because (-1)^2 + (-1) + (-1)*(-1) + (-1) + (-1)^2 = 1.
4 is a term because 2^2 + 2 + 2*(-2) + (-2) + (-2)^2 = 4.
24 is a term because 2^2 + 2 + 2*3 + 3 + 3^2 = 24.

Michael De Vlieger, Table of n, a(n) for n = 1..10583
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Hilbert Space Fragmentation in the Chiral Luttinger Liquid, arXiv:2409.10359 [cond-mat.str-el], 2024. See pp. 5, 8.
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Momentum-space modulated symmetries in the Luttinger liquid, Phys. Rev. B (2025) Vol. 111, Art. No. 165105. See p. 4.

Cf. A000290, A002378, A003136, A045944, A202822.

f[{i_, j_}] := (i^2 + i*j + j^2 + i + j); Union@ Map[f, Tuples[Range[-10, 10], 2] ] (* Michael De Vlieger, Sep 23 2024, after Harvey P. Dale at A202822 *)

x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%3==1, print1((n-1)/3, ", ")));

is(n) = sumdiv( n, d, kronecker( -3, d));
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

a(n) = (A202822(n) - 1) / 3.

A266836 Odd Löschian numbers.

Original entry on oeis.org

1, 3, 7, 9, 13, 19, 21, 25, 27, 31, 37, 39, 43, 49, 57, 61, 63, 67, 73, 75, 79, 81, 91, 93, 97, 103, 109, 111, 117, 121, 127, 129, 133, 139, 147, 151, 157, 163, 169, 171, 175, 181, 183, 189, 193, 199, 201, 211, 217, 219, 223, 225, 229, 237, 241, 243, 247, 259, 271, 273, 277, 279, 283, 289, 291, 301, 307, 309

Offset: 1

Joerg Arndt, Jan 04 2016

nonn

Löschian numbers are numbers of the form x^2 + xy + y^2 for integers x, y; they can all be written in the form 4^e * m where e is a nonnegative integer and m is an odd Löschian number. - Charles R Greathouse IV, Jan 04 2016

Charles R Greathouse IV, 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.

Cf. Loeschian numbers: A003136 (all), A202822 (3*k+1), A260682 (6*k+1).

fQ[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x y + y^2, {x, y}, Integers]]]; Select[2 Range@ 155 - 1, fQ] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003136 *)

is(n)=(n%2==1) && #bnfisintnorm(bnfinit(z^2+z+1), n);

x='x+O('x^1000); p=eta(x)^3/eta(x^3); for(n=0, 999, if(polcoeff(p, n) != 0 && n%2==1, print1(n, ", "))) \\ Altug Alkan, Jan 04 2016

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, if(t%2, listput(v, t)); y++)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

A270248 Even Löschian numbers.

Original entry on oeis.org

0, 4, 12, 16, 28, 36, 48, 52, 64, 76, 84, 100, 108, 112, 124, 144, 148, 156, 172, 192, 196, 208, 228, 244, 252, 256, 268, 292, 300, 304, 316, 324, 336, 364, 372, 388, 400, 412, 432, 436, 444, 448, 468, 484, 496, 508, 516, 532, 556, 576, 588, 592, 604, 624, 628, 652, 676

Offset: 1

Altug Alkan, Mar 14 2016

nonn

Even numbers of the form x^2 - xy + y^2.

			Even number 12 is a term because 12 = 2^2 + 2*2 + 2^2.

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

Cf. Loeschian numbers: A003136 (all), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).

Select[Range[0, 680, 2], Resolve@ Exists[{x, y}, Reduce[# == (x^2 - x y + y^2), {x, y}, Integers]] &] (* Michael De Vlieger, Mar 15 2016 *)

x='x+O('x^800); p=eta(x)^3/eta(x^3); for(n=0, 799, if(polcoeff(p, n) != 0 && n % 2 == 0, print1(n, ", ")));

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

a(n) = 2 * A270050(n) = 4 * A003136(n).

A338992 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at the center of one hexagon.

Original entry on oeis.org

0, 6, 6, 12, 12, 6, 12, 6, 12, 12, 12, 12, 18, 12, 12, 6, 12, 12, 12, 12, 24, 12, 6, 12, 12, 12, 6, 12, 12, 24, 12, 12, 12, 12, 12, 18, 12, 12, 12, 12, 18, 12, 12, 12, 24, 12, 12, 12, 12, 24, 6, 24, 12, 12, 12, 12, 6, 12, 24, 12, 12, 12, 12, 12, 12, 12, 24, 12, 18

Offset: 0

Szymon Lukaszyk, Nov 17 2020

nonn

Radii of these circles are square roots of A202822.

This is A217219 with zeros dropped, except for a(0). - Andrey Zabolotskiy, Jun 21 2022

Szymon Lukaszyk, MATLAB script
Szymon Lukaszyk, A short note about the geometry of graphene, ResearchGate, (2020).
Szymon Lukaszyk, Illustration of the subsequent 36 radii around a hex center.

Cf. A202822, A217219, A338947 (similar but with circles centered at a vertex of one hexagon).

A266918 Perfect power Löschian numbers.

Original entry on oeis.org

1, 4, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2304, 2401, 2500

Offset: 1

Altug Alkan, Jan 06 2016

nonn

Inspired by A266836. See first comment in A266836.
Intersection of A001597 and A003136.
Obviously, this sequence contains all positive squares.
Perfect powers that are not the Löschian numbers are 8, 32, 125, 128, 216, 512, 1000, 1331, 2048, 2744, 3125, 3375, 4913, 5832, 7776, ...

			25 is a term because 25 = 5^2 = 5^2 + 5*0 + 0^2.
27 is a term because 27 = 3^3 = 3^2 + 3*3 + 3^2.
243 is a term because 243 = 3^5 = 9^2 + 9*9 + 9^2.
343 is a term because 343 = 7^3 = 18^2 + 18*1 + 1^2.

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

Cf. Loeschian numbers: A003136 (all), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).

Cf. A001597.

fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; gQ[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x y + y^2, {x, y}, Integers]]]; Select[Range@ 2500, fQ@# && gQ@# &] (* Michael De Vlieger, Jan 06 2016, after Ant King at A001597 and Jean-François Alcover at A003136 *)

x='x+O('x^10^4); p=eta(x)^3/eta(x^3); for(n=0, 9999, if(polcoeff(p, n) != 0 && (ispower(n) || n==1), print1(n, ", ")));

is(n) = (ispower(n) || n==1) && #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e4, if(is(n), print1(n, ", ")));

A270672 Löschian numbers (A003136) that are multiples of 3.

Original entry on oeis.org

0, 3, 9, 12, 21, 27, 36, 39, 48, 57, 63, 75, 81, 84, 93, 108, 111, 117, 129, 144, 147, 156, 171, 183, 189, 192, 201, 219, 225, 228, 237, 243, 252, 273, 279, 291, 300, 309, 324, 327, 333, 336, 351, 363, 372, 381, 387, 399, 417, 432, 441, 444, 453, 468, 471, 489, 507, 513, 516, 525, 543, 549

Offset: 1

Altug Alkan, Mar 21 2016

nonn

Numbers of the form 3*(x^2 + xy + y^2).

Intersection of A008585 and A003136.

			21 is a term because 21 = 3*7 = 4^2 + 4*1 + 1^2.

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

Cf. Loeschian numbers: A003136 (all), A270248 (2*k), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).

Select[Range[0, 600], Resolve[Exists[{x, y}, Reduce[# == 3 (x^2 + x y + y^2), {x, y}, Integers]]] &] (* Michael De Vlieger, Mar 21 2016 *)

x='x+O('x^1000); p=eta(x)^3/eta(x^3); for(n=0, 999, if(polcoeff(p, n) != 0 && n % 3 == 0, print1(n, ", ")));

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+=3)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

A261558 Euclid numbers (A006862) of the form 3(ii + ij + jj + i + j) + 1 where i and j are integers.

Original entry on oeis.org

7, 31, 211, 2311, 510511, 6469693231, 200560490131, 304250263527211, 117288381359406970983271, 7858321551080267055879091, 40729680599249024150621323471, 232862364358497360900063316880507363071, 279734996817854936178276161872067809674997231

Offset: 1

Altug Alkan, Nov 18 2015

nonn

Intersection of A006862 and A202822.

			a(1) = 7 because 7 = 2*3 + 1 = 3*(1^2 + 1*0 + 0^2 + 1 + 0) + 1.

Eric Weisstein's World of Mathematics, Euclid Number

Cf. A003136, A006862, A202822.

a(n) = prod(k=1, n, prime(k)) + 1;
isA(n) = if( n<1 || (n%3 == 0), 0, 0 != sumdiv( n, d, kronecker( -3, d)));
for(n=0, 30, if(isA(a(n)), print1(a(n), ", ")))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Julia

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Python

Formula

A260682 Löschian numbers (A003136) of the form 6*k+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

PARI

A267137 Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A266836 Odd Löschian numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

A270248 Even Löschian numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A338992 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at the center of one hexagon.

Views

Author

Keywords

Comments

A261558 Euclid numbers (A006862) of the form 3(ii + ij + jj + i + j) + 1 where i and j are integers.