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

%I A024606 #54 Feb 16 2025 08:32:34
%S A024606 7,13,19,21,28,31,37,39,43,49,52,57,61,63,67,73,76,79,84,91,93,97,103,
%T A024606 109,111,112,117,124,127,129,133,139,147,148,151,156,157,163,169,171,
%U A024606 172,175,181,183,189,193,196,199,201,208,211,217,219,223,228,229,237,241,244,247
%N A024606 Numbers of form x^2 + xy + y^2 with distinct x and y > 0.
%C A024606 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
%C A024606 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
%C A024606 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
%C A024606 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
%C A024606 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
%H A024606 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H A024606 Michael Somos, <a href="/A010815/a010815.txt">A Multisection of q-Series</a>
%H A024606 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A024606 <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F A024606 A004016(a(n)) >= 12. - _Jean-Christophe Hervé_, Nov 24 2013
%e A024606 a(1) = 7 = 1^2 + 2 + 2^2.
%t A024606 Take[Union[Flatten[Table[x^2 + x*y + y^2, {x, 15}, {y, x - 1}]]], 60] (* _Robert G. Wilson v_, Nov 24 2013 *)
%o A024606 (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
%Y A024606 Cf. A003136, A004016, A024614, A074628, A088534, A118886, A232436.
%K A024606 nonn,easy
%O A024606 1,1
%A A024606 _Clark Kimberling_
%E A024606 Definition modified by _Alonso del Arte_ and _Jean-Christophe Hervé_, Nov 25 2013