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A154777 Numbers of the form x^2 + 2*y^2 with positive integers x and y. 37

%I

%S 3,6,9,11,12,17,18,19,22,24,27,33,34,36,38,41,43,44,48,51,54,57,59,66,

%T 67,68,72,73,75,76,81,82,83,86,88,89,96,97,99,102,107,108,113,114,118,

%U 121,123,129,131,132,134,136,137,139,144,146,147,150,152,153,162,163

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

%C 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).

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

%C 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

%C If m is a term, 2*m is also. - _Zak Seidov_, Nov 30 2011

%C 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

%C This sequence is closed under multiplication because A002479 is. -__Jerzy R Borysowicz_, Jun 13 2020

%H Zak Seidov, <a href="/A154777/b154777.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e a(2) = 6 = 2^2 + 2*1^2 is the second smallest number that can be written in this way.

%t 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 *)

%o (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))}

%o for( n=1,200, isA154777(n) & print1(n","))

%Y Cf. A000408, A002479, A155560, A338432 (triangle version of array), A339047 (multiplicities).

%K easy,nonn

%O 1,1

%A _M. F. Hasler_, Jan 24 2009

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Last modified June 18 04:41 EDT 2021. Contains 345098 sequences. (Running on oeis4.)