cp's OEIS Frontend

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

%I A154777 #36 Jan 21 2025 11:24:51
%S A154777 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 A154777 67,68,72,73,75,76,81,82,83,86,88,89,96,97,99,102,107,108,113,114,118,
%U A154777 121,123,129,131,132,134,136,137,139,144,146,147,150,152,153,162,163
%N A154777 Numbers of the form x^2 + 2*y^2 with positive integers x and y.
%C A154777 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 A154777 Also, subsequence of A000408 (with 2*y^2 = y^2 + z^2).
%C A154777 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 A154777 If m is a term, 2*m is also. - _Zak Seidov_, Nov 30 2011
%C A154777 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 A154777 This sequence is closed under multiplication because A002479 is. - _Jerzy R Borysowicz_, Jun 13 2020
%H A154777 Zak Seidov, <a href="/A154777/b154777.txt">Table of n, a(n) for n = 1..10000</a>
%e A154777 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 A154777 a(2) = 6 = 2^2 + 2*1^2 is the second smallest number that can be written in this way.
%t A154777 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 A154777 (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 A154777 for( n=1,200, isA154777(n) & print1(n","))
%Y A154777 Subsequence of A002479 and hence of A000408.
%Y A154777 Cf. A155560, A338432 (triangle version of array), A339047 (multiplicities).
%K A154777 easy,nonn
%O A154777 1,1
%A A154777 _M. F. Hasler_, Jan 24 2009