A088534 -id:A088534 - cp's OEIS frontend

A326421 Least numbers k such that A088534(k) is nondecreasing.

0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 637, 931, 1183, 1519, 1729, 2821, 3367

Offset: 1

Hugo Pfoertner, Sep 12 2019

nonn

Least squared radius of a circle around a grid point of the hexagonal lattice such that the number of grid points on this circle is not smaller than the number of grid points on any circle around a grid point of the hexagonal lattice with smaller radius. a(1) = 0 by convention.

Hugo Pfoertner, Table of n, a(n) for n = 1..163

Cf. A003136, A088534, A118886, A230655, A326311.

\\ after M. F. Hasler's A088534(n,d)
r2=0;for(k=0,3500,my(a088534=sum(x=0, sqrt(k\3), sum(y=max(x, sqrtint(k-x^2)\2), sqrtint(k-2*x^2), x^2+x*y+y^2==k)));if(a088534>=r2,r2=a088534;print1(k,", ")))

A003627 Primes of the form 3n-1.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane and Mira Bernstein

Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003
A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - Juri-Stepan Gerasimov, Jan 28 2010
Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
2 adjoined to A007528. - David A. Corneth, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
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..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Eisenstein Prime
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317.
Index to sequences related to decomposition of primes in quadratic fields

Primes of form 3n+1 give A002476.
These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.
Cf. A001359, A007528, A007645, A221717, A057145.
Subsequence of A034020.

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011

t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644.
Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

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)

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613

Offset: 1

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v

Also, odd primes p such that -3 is a square mod p. - N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3 - y^3)/(x - y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (-1+sqrt(-3))/2. The present sequence gives the rational primes which are also Eisenstein primes. - N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633. - T. D. Noe, May 19 2008
Conjecture: this sequence is Union(A002383,A162471). - Daniel Tisdale, Jul 04 2009
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p - 1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p). - Jon Perry, Feb 02 2014
The prime factors of A002061. - Richard R. Forberg, Dec 10 2014
This sequence gives the primes p which solve s^2 == -3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = -3). p = 2 is not a solution. x^2 == -3 (mod 4) has solutions for all odd x. x^2 == -3 (mod  p) has for odd primes p, not 3, the solutions of Legendre(-3|p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == -3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)). - Wolfdieter Lang, May 22 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

T. D. Noe, Table of n, a(n) for n = 1..1000
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Eisenstein Integer.

Subsequence of A003136.
Subsequences include A002407, A002648, and A201477.
Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).
Cf. A000040, A003627.
Primes in A003136 and A034017.

a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

select(isprime,[3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014

Join[{3},Select[Prime[Range[150]],Mod[#,3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)

forprime(p=2,1e3,if(p%3<2,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

p == 0 or 1 (mod 3).
{3} UNION A002476. - R. J. Mathar, Oct 28 2008
A007645 UNION A003627 = A000040. - Juri-Stepan Gerasimov, Jan 28 2010

Entry revised by N. J. A. Sloane, Jan 29 2013

A034020 Not of the form x^2 + x*y + y^2.

Original entry on oeis.org

2, 5, 6, 8, 10, 11, 14, 15, 17, 18, 20, 22, 23, 24, 26, 29, 30, 32, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 50, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 72, 74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105

Offset: 1

N. J. A. Sloane

nonn

Appears to be the sequence of nonsquare n such that sigma(n)==0 (mod 3). - Benoit Cloitre, Sep 17 2002
First counterexample is 147 = 11^2 + 11*2 + 2^2 since sigma(147) = 3 * 76. See A087943. - Charles R Greathouse IV, Jun 29 2011
Numbers n such that n-th coefficient of eta(x)^3/eta(x^3) is zero where eta(x) coefficients are given by A010815. - Benoit Cloitre, Oct 06 2005
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003136 (complement), A003627 (primes). Subsequence of A087943.

a034020 n = a034020_list !! (n-1)
a034020_list = f [0..] a003136_list where
   f (x:xs) ys'@(y:ys) | x < y = x : f xs ys'
                       | otherwise = f xs ys
-- Reinhard Zumkeller, Oct 30 2011

Select[Range@ 105, ! Resolve[Exists[{x, y}, Reduce[# == x^2 + x y + y^2, {x, y}, Integers]]] &] (* Michael De Vlieger, Jan 06 2016 *)

default(seriesprecision, 105); for(n=1, 105, if (polcoeff(eta(x)^3/eta(x^3)+O(x^(n+1)), n) == 0, print1(n,","))) \\ Benoit Cloitre, Oct 06 2005

x='x+O('x^100); p=eta(x)^3/eta(x^3); for(n=1, 99, if(polcoeff(p, n)==0, print1(n, ", "))); \\ Altug Alkan, Nov 08 2015

list(lim)=my(v=List(), y, t); lim\=1; for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, t); y++)); v=Set(v); setminus([2..lim], v) \\ Charles R Greathouse IV, Jul 05 2017

a(n) ~ n. - Charles R Greathouse IV, Jul 05 2017

More terms from James Sellers, May 04 2000

Correct offset=1 by Ray Chandler, Jan 29 2009

A118886 Numbers expressible as x^2 + x*y + y^2, 0 <= x <= y, in 2 or more ways.

Original entry on oeis.org

49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 637, 651, 676, 679, 703, 721, 741, 763, 777, 784, 793, 817, 819, 868, 871, 889, 903, 931, 949, 961, 973, 988, 1027, 1029, 1036, 1057, 1083, 1092, 1099

Offset: 1

Franklin T. Adams-Watters, May 03 2006

nonn

Squares of distances between two points in the triangular lattice in two or more nontrivially different ways.
Numbers whose prime factorization contains at least two (not necessarily distinct) primes congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. Products of two elements of A024606.
If k is in the sequence then so is k * m^2 for m > 0. - David A. Corneth, Jun 21 2018

			a(2) = 91 = 1^2 + 1*9 + 9^2 = 5^2 + 5*6 + 6^2;
a(45) = 931 = 1^2+1*30+30^2 = 9^2+9*25+25^2 = 14^2+14*21+21^2;
a(97) = 1729 = 3^2+3*40+40^2 = 8^2+8*37+37^2 = 15^2+15*32+32^2 = 23^2+23*25+25^2. - _Reinhard Zumkeller_, Oct 30 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Mazel, I. Stuhl, Y. Suhov, Hard-core configurations on a triangular lattice and Eisenstein primes, arXiv:1803.04041 [math.PR], 2018.

Subsequence of Loeschian numbers A003136.
Complement of A198772 with respect to A003136.
Subsequences: A198773, A198774, A198775.
Cf. A024606, A118882.

a118886 n = a118886_list !! (n-1)
a118886_list = filter ((> 1) . a088534) a003136_list
-- Reinhard Zumkeller, Oct 30 2011

function isA118886(n)
    n < 49 && return false
    n % 3 == 2 && return false
    M = Int(round(2*sqrt(n/3)))
    count = 0
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && (count += 1)
        count == 2 && break
    end
    return count >= 2
end
A118886list(upto) = [n for n in 0:upto if isA118886(n)]
A118886list(1099) |> println # Peter Luschny, Mar 17 2018

amax = 2000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
A118886 = Select[Range[0, 3 xmax^2], # <= amax && f[#] > 1&] (* Jean-François Alcover, Jun 21 2018 *)

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

A088534(a(n)) > 1. - Reinhard Zumkeller, Oct 30 2011

A198772 Numbers having exactly one representation by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

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, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 139, 144, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 192, 193

Offset: 1

Reinhard Zumkeller, Oct 30 2011

nonn

			a(20) = 48 = 4^2 + 4*4 + 4^2, A088534(48) = 1;
a(21) = 52 = 2^2 + 2*6 + 6^2, A088534(52) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of Loeschian numbers A003136.
Complement of A118886 with respect to A003136.
Cf. A198773, A198774, A198775.

a198772 n = a198772_list !! (n-1)
a198772_list = filter ((== 1) . a088534) a003136_list

function isA198772(n)
    M = Int(round(2*sqrt(n/3)))
    count = 0
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && (count += 1)
        count == 2 && break
    end
    return count == 1
end
A198772list(upto) = [n for n in 0:upto if isA198772(n)]
A198772list(193) |> println # Peter Luschny, Mar 17 2018

amax = 200; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
A198772 = Select[Range[0, 3 xmax^2], # <= amax && f[#] == 1&] (* Jean-François Alcover, Jun 21 2018 *)

A088534(a(n)) = 1.

a(n) = A034022(n) for n <= 32.

A024606 Numbers of form x^2 + xy + y^2 with distinct x and y > 0.

Original entry on oeis.org

7, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 63, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247

Offset: 1

Clark Kimberling

Alternatively, numbers expressible in more than one way as i^2 - ij + j^2, where 1 <= i <= j or 1 <= i < j. The following argument shows that the conditions i <= j or i < j are here equivalent. Note first that i^2 - ij + j^2 = (j - i)^2 - (j - i)*j + j^2, so the only non-duplicated values i^2 - ij + j^2 with 1 <= i < j are when j = 2i, whence i^2 - ij + j^2 = 3i^2. On the other hand, the values with i = j are j^2. There are no integer solutions to 3i^2 = j^2 with i >= 1. - Franklin T. Adams-Watters, May 03 2006
Numbers whose prime factorization contains at least one prime congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. - Franklin T. Adams-Watters, May 03 2006
This is a subsequence of Loeschian numbers A003136, closed under multiplication. Its primitive elements are those with exactly one prime factor of form 6k + 1 with multiplicity one (A232436). - Jean-Christophe Hervé, Nov 22 2013
a(1)*a(2)*a(3) = 1729, the Hardy-Ramanujan taxicab number. 1729 is then in the sequence, by the argument of the preceding comment. - Jean-Christophe Hervé, Nov 24 2013
1729 is also the least term that can be written in 4 distinct ways in the given form. Sequence A024614 does not include the restriction x != y, it is the disjoint union of this sequence and A033428 (i.e., 3*x^2) (without 0). - M. F. Hasler, Mar 05 2018

			a(1) = 7 = 1^2 + 2 + 2^2.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, A Multisection of q-Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003136, A004016, A024614, A074628, A088534, A118886, A232436.

Take[Union[Flatten[Table[x^2 + x*y + y^2, {x, 15}, {y, x - 1}]]], 60] (* Robert G. Wilson v, Nov 24 2013 *)

for(k=1,247,my(a088534=sum(x=0,sqrt(k\3),sum(y=max(x,sqrtint(k-x^2)\2),sqrtint(k-2*x^2),x^2+x*y+y^2==k)),a004016d6=sumdiv(k,d,(d%3==1)-(d%3==2)));if(a088534!=a004016d6,print1(k,", "))) \\ Hugo Pfoertner, Sep 22 2019

A004016(a(n)) >= 12. - Jean-Christophe Hervé, Nov 24 2013

Definition modified by Alonso del Arte and Jean-Christophe Hervé, Nov 25 2013

A198773 Numbers having exactly two representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.

Original entry on oeis.org

49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 651, 676, 679, 703, 721, 741, 763, 777, 784, 793, 817, 819, 868, 871, 889, 903, 949, 961, 973, 988, 1027, 1029, 1036, 1057, 1083, 1092, 1099, 1141, 1147

Offset: 1

Reinhard Zumkeller, Oct 30 2011

nonn

A088534(a(n)) = 2; subsequence of A118886, see also A003136.

			a(10) = 273 = 1^2 + 1*16 + 16^2 = 8^2 + 8*11 + 11^2, A088534(273) = 2;
a(11) = 301 = 4^2 + 4*15 + 15^2 = 9^2 + 9*11 + 11^2, A088534(301) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A198772, A198774, A198775.

a198773 n = a198773_list !! (n-1)
a198773_list = filter ((== 2) . a088534) a003136_list

amax = 2000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
A198773 = Select[Range[0, 3 xmax^2], # <= amax && f[#] == 2&] (* Jean-François Alcover, Jun 21 2018 *)

is(n)=my(t=#bnfisintnorm(bnfinit(z^2+z+1), n)); t==3 || t==4;
select(n->is(n), vector(1500,j,j)) \\ Joerg Arndt, Jan 11 2015

A198774 Numbers having exactly three representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.

Original entry on oeis.org

637, 931, 1183, 1519, 1813, 1911, 2107, 2401, 2527, 2548, 2793, 2989, 3211, 3283, 3549, 3577, 3724, 3871, 4557, 4693, 4732, 4753, 5047, 5239, 5341, 5439, 5733, 6076, 6223, 6253, 6321, 6727, 6811, 7203, 7252, 7267, 7399, 7581, 7644, 7693, 7987, 8379, 8428

Offset: 1

Reinhard Zumkeller, Oct 30 2011

nonn

A088534(a(n)) = 3; subsequence of A118886, see also A003136.

			a(1) = 637 = 4^2+4*23+23^2 = 7^2+7*21+21^2 = 12^2+12*17+17^2, A088534(637)=3;
a(2) = 931 = 1^2+1*30+30^2 = 9^2+9*25+25^2 = 14^2+14*21+21^2, A088534(273)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A198772, A198773, A198775.

a198774 n = a198774_list !! (n-1)
a198774_list = filter ((== 3) . a088534) a003136_list

amax = 10000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
A198774 = Select[Range[0, 3 xmax^2], # <= amax && f[#] == 3&] (* Jean-François Alcover, Jun 21 2018 *)

cp's OEIS Frontend

A326421 Least numbers k such that A088534(k) is nondecreasing.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A003627 Primes of the form 3n-1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

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

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A034020 Not of the form x^2 + x*y + y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

Formula

Extensions

A118886 Numbers expressible as x^2 + x*y + y^2, 0 <= x <= y, in 2 or more ways.

Views

Author

Keywords