OFFSET
1,1
COMMENTS
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.
Sequence contains every even number >=4 and no odd numbers.
EXAMPLE
a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.
MATHEMATICA
nn = 80; s = Select[Range[4 nn], Length[PowersRepresentations[#, 2,
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 *)
PROG
(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
findk(n) = {for (k=1, n, if (issquare(n+k) && issquare(n-k), return (k)); ); }
lista(nn) = {for (n=1, nn, if (issum(n), print1(findk(n), ", "); ); ); } \\ Michel Marcus, Mar 31 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Bob Selcoe, Mar 23 2016
STATUS
approved