

A024606


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


8



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
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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 nonduplicated 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. AdamsWatters, 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. AdamsWatters, 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).  JeanChristophe Hervé, Nov 22 2013
a(1)*a(2)*a(3) = 1729, the HardyRamanujan taxicab number. 1729 is then in the sequence, by the argument of the preceding comment.  JeanChristophe 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

Table of n, a(n) for n=1..60.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, A Multisection of qSeries
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice


FORMULA

A004016(a(n)) >= 12.  JeanChristophe 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(kx^2)\2), sqrtint(k2*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

Cf. A003136, A004016, A024614, A074628, A088534, A118886, A232436.
Sequence in context: A330946 A160007 A024613 * A074628 A232436 A274437
Adjacent sequences: A024603 A024604 A024605 * A024607 A024608 A024609


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling


EXTENSIONS

Definition modified by Alonso del Arte and JeanChristophe Hervé, Nov 25 2013


STATUS

approved



