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A270835 a(n) = smallest k such that A004431(n) +/- k are both positive squares.

%I A270835 #30 Apr 19 2016 11:10:32
%S A270835 4,6,12,8,16,24,10,20,30,12,24,40,36,14,48,28,42,60,16,32,48,70,64,18,
%T A270835 36,80,54,72,96,20,40,90,60,112,80,108,22,44,66,120,88,24,110,48,140,
%U A270835 72,132,96,160,120,26,52,78,144,180
%N A270835 a(n) = smallest k such that A004431(n) +/- k are both positive squares.
%C A270835 There can be more than one value k such that A004431(n) +/- k are both positive squares; i.e., when there are multiple ways to express A004431(n) as the sum of positive squares. These are the terms which appear more than once in A055096. For example A004431(19) = 65 = {(1^2 + 8^2), (4^2 + 7^2)}: 65 +/- 16 = {7^2, 9^2} and 65 +/- 56 = {3^2, 11^2}. So a(19) = 16 rather than 56.
%C A270835 Sequence contains every even number >=4 and no odd numbers.
%e A270835 a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.
%t A270835 nn = 80; s = Select[Range[4 nn], Length[PowersRepresentations[#, 2,
%t A270835 2] /. {{0, _} -> Nothing, {a_, b_} /; a == b -> Nothing}] > 0 &]; Table[SelectFirst[Range[10 nn], And[IntegerQ@ Sqrt[s[[n]] + #], IntegerQ@ Sqrt[s[[n]] - #]] &], {n, nn}] (* _Michael De Vlieger_, Mar 24 2016, Version 10 *)
%o A270835 (PARI) issum(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1; \\ after A004431
%o A270835 findk(n) = {for (k=1, n, if (issquare(n+k) && issquare(n-k), return (k)););}
%o A270835 lista(nn) = {for (n=1, nn, if (issum(n), print1(findk(n), ", ");););} \\ _Michel Marcus_, Mar 31 2016
%Y A270835 Cf. A001844, A004431, A055096.
%K A270835 nonn
%O A270835 1,1
%A A270835 _Bob Selcoe_, Mar 23 2016