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 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 (list; graph; refs; listen; history; text; internal format)
 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 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 EXTENSIONS Definition modified by Alonso del Arte and Jean-Christophe Hervé, Nov 25 2013 STATUS approved

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Last modified March 23 23:09 EDT 2023. Contains 361454 sequences. (Running on oeis4.)