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

1,1

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.

Links

Table of n, a(n) for n=1..55.

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

Cf. A001844, A004431, A055096.

Sequence in context: A293836 A278252 A274298 * A272063 A152678 A110758

Adjacent sequences: A270832 A270833 A270834 * A270836 A270837 A270838

Keyword

nonn

Author

Bob Selcoe, Mar 23 2016

Status

approved