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A303234 Numbers of the form x*(x+1)/2 + 2^y with x and y nonnegative integers.

%I A303234 #12 Mar 11 2023 08:05:47
%S A303234 1,2,3,4,5,7,8,9,10,11,12,14,16,17,18,19,22,23,25,26,29,30,31,32,33,
%T A303234 35,36,37,38,40,42,44,46,47,49,52,53,56,57,59,60,61,63,64,65,67,68,70,
%U A303234 71,74,77,79,80
%N A303234 Numbers of the form x*(x+1)/2 + 2^y with x and y nonnegative integers.
%C A303234 Conjecture: Any integer n > 1 can be written as the sum of two terms of the current sequence.
%C A303234 This is equivalent to the author's conjecture in A303233.
%H A303234 Zhi-Wei Sun, <a href="/A303234/b303234.txt">Table of n, a(n) for n = 1..10000</a>
%H A303234 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H A303234 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
%H A303234 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e A303234 a(1) = 1 with 1 = 0*(0+1)/2 + 2^0.
%e A303234 a(2) = 2 with 2 = 1*(1+1)/2 + 2^0 = 0*(0+1)/2 + 2^1.
%t A303234 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A303234 tab={};Do[Do[If[SQ[8(n-2^k)+1],tab=Append[tab,n];Goto[aa]],{k,0,Log[2,n]}];Label[aa],{n,1,80}];Print[tab]
%Y A303234 Cf. A000079, A000217, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233.
%K A303234 nonn
%O A303234 1,2
%A A303234 _Zhi-Wei Sun_, Apr 20 2018