OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

EXTENSIONS

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

STATUS

approved