A073412 Lesser of three consecutive nonsquare integers each of which is the sum of two squares.

72, 232, 520, 584, 800, 808, 1096, 1152, 1312, 1664, 1744, 1800, 1872, 1960, 2248, 2312, 2384, 2592, 2824, 3328, 3392, 3528, 4112, 4176, 4328, 5120, 5408, 5904, 6056, 6120, 6272, 6352, 6408, 6568, 6920, 8080, 8144, 8296, 8352, 8584, 8712, 9160, 9376

Offset: 1

Author

Jason Earls, Aug 23 2002

Keywords

• nonn

Comments

a(n) == 0 mod 8. - Zak Seidov, Jan 26 2013

Is this sequence the same as A064715? - Zak Seidov, Jan 26 2013

This sequence is distinct from A064715 since it allows numbers equal to twice a square, like 72, 1152, 2592, 3528, etc. - Giovanni Resta, Jan 29 2013

This sequence lists lesser of three consecutive nonsquare integers each of which is the sum of two squares. So this sequence is a subsequence of A064716. - Altug Alkan, Jul 07 2016

Examples

			232 is here since 232 = 6^2 + 14^2; 233 = 8^2 + 13^2; 234 = 3^2 + 15^2 and 232, 233, 234 are all nonsquares.
288 is not a term because 288 = 12^2 + 12^2, 289 = 8^2 + 15^2, 290 = 1^2 + 17^2 but 289 is also square.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000404,A000415, A001481, A064716.

Programs

Maple

is415:= proc(n) local F;
  if issqr(n) then return false fi;
  F:= select(t -> t[1] mod 4 = 3, ifactors(n)[2]);
  andmap(t -> t[2]::even, F);
end proc:
Q:= select(is415, [seq(seq(8*i+j,j=0..2),i=1..2000)]):
Q[select(t -> Q[t+2]-Q[t]=2, [$1..nops(Q)-2])]; # Robert Israel, Mar 05 2018

Mathematica

nsQ[x_] := !IntegerQ[Sqrt[x]];
prQ[x_] := With[{pr = PowersRepresentations[x, 2, 2]}, pr != {} && AllTrue[pr[[1]], IntegerQ]];
selQ[x_] := nsQ[x] && nsQ[x+1] && nsQ[x+2] && prQ[x] && prQ[x+1] && prQ[x+2];
Select[8 Range[10000], selQ] (* Jean-François Alcover, Jun 11 2020 *)

PARI

isA001481(n) = my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
isok(n) = isA001481(n) && isA001481(n+1) && isA001481(n+2) && !issquare(n) && !issquare(n+1);
lista(nn) = for(n=1, nn, if(isok(8*n), print1(8*n, ", "))); \\ Altug Alkan, Jul 07 2016

Extensions

Edited by Robert Israel, Mar 05 2018