A154777 Numbers of the form x^2 + 2*y^2 with positive integers x and y.

3, 6, 9, 11, 12, 17, 18, 19, 22, 24, 27, 33, 34, 36, 38, 41, 43, 44, 48, 51, 54, 57, 59, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 99, 102, 107, 108, 113, 114, 118, 121, 123, 129, 131, 132, 134, 136, 137, 139, 144, 146, 147, 150, 152, 153, 162, 163

Offset: 1

Author

M. F. Hasler, Jan 24 2009

Keywords

• easy
• nonn

Comments

Subsequence of A002479 (which allows for x=0 and/or y=0). See there for further references. See A155560 cf for intersection of sequences of type (x^2 + k*y^2).

Also, subsequence of A000408 (with 2*y^2 = y^2 + z^2).

If m and n are terms also n*m is (in particular any power of term is also a term). - Zak Seidov, Nov 30 2011

If m is a term, 2*m is also. - Zak Seidov, Nov 30 2011

Select terms that are multiples of 25: 75, 150, 225, 275, 300, 425, 450, 475, 550, 600, 675, 825, 850, 900, 950, 1025, 1075, 1100, ... Divide them by 25: 3, 6, 9, 11, 12, 17, 18, 19, 22, 24, 27, 33, 34, 36, 38, 41, 43, 44, 48, 51, 54, 57, 59, 66, 67, 68, 72, ... and we get the original sequence. - Zak Seidov, Dec 01 2011

This sequence is closed under multiplication because A002479 is. - Jerzy R Borysowicz, Jun 13 2020

Examples

			a(1) = 3 = 1^2 + 2*1^2 is the least number that can be written as A + 2B where A, B are positive squares.
a(2) = 6 = 2^2 + 2*1^2 is the second smallest number that can be written in this way.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A002479 and hence of A000408.

Cf. A155560, A338432 (triangle version of array), A339047 (multiplicities).

Programs

Mathematica

f[upto_]:=Module[{max=Ceiling[Sqrt[upto-1]]},Select[Union[ First[#]^2+ 2Last[#]^2&/@Tuples[Range[13],{2}]],#<=upto&]]; f[200] (* Harvey P. Dale, Jun 17 2011 *)

PARI

isA154777(n,/* use optional 2nd arg to get other analogous sequences */c=2) = { for( b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,200, isA154777(n) & print1(n","))