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

4, 6, 12, 8, 16, 24, 10, 20, 30, 12, 24, 40, 36, 14, 48, 28, 42, 60, 16, 32, 48, 70, 64, 18, 36, 80, 54, 72, 96, 20, 40, 90, 60, 112, 80, 108, 22, 44, 66, 120, 88, 24, 110, 48, 140, 72, 132, 96, 160, 120, 26, 52, 78, 144, 180

Offset: 1

Author

Bob Selcoe, Mar 23 2016

Keywords

• nonn

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.

Examples

			a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.
		

Crossrefs

Cf. A001844, A004431, A055096.

Programs

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 *)

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