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%I #19 Nov 24 2020 03:25:45
%S 7,28,31,39,63,79,103,111,112,124,127,151,156,175,183,199,223,247,252,
%T 271,279,316,327,343,351,367,399,412,439,444,448,463,471,487,496,508,
%U 511,543,559,567,604,607,624,631,679,687,700,703,711,727,732,751,775,796,823,831,847,871
%N Numbers of the form x^2 + xy + y^2 (A003136) that are not of the form a^2 + b^2 + c^2 where x, y, a, b and c are integers.
%C Intersection of A003136 and A004215.
%C Motivation for this sequence is the equation x^2 + x*y + y^2 = a^2 + b^2 + c^2 where x, y, a, b and c are integers.
%C Löschian numbers of the form a^2 + b^2 + c^2, where a, b and c are integers, are 0, 1, 3, 4, 9, 12, 13, 16, 19, 21, 25, 27, 36, 37, 43, 48, 49, 52, 57, 61, 64, ...
%H Amiram Eldar, <a href="/A267432/b267432.txt">Table of n, a(n) for n = 1..10000</a>
%e 7 is a term because it is a Löschian number and there is no integer values of a, b and c for the equation 7 = a^2 + b^2 + c^2.
%e Löschian number 19 is not a term because 19 = 5^2 + 5*(-2) + (-2)^2 = 1^2 + 3^2 + 3^2.
%t Select[Range@ 900, And[Resolve[Exists[{x, y}, Reduce[# == x^2 + x y + y^2, {x, y}, Integers]]], !Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]]] &] (* _Michael De Vlieger_, Jan 15 2016, after _Jean-François Alcover_ at A003136 *)
%o (PARI) isA004215(n) = {my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } {for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
%o isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
%o for(n=0, 1e3, if(isA004215(n) && isA003136(n), print1(n, ", ")));
%Y Cf. A003136, A004215.
%K nonn
%O 1,1
%A _Altug Alkan_, Jan 15 2016
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