cp's OEIS Frontend

A303393 Numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.

%I A303393 #5 Apr 23 2018 04:04:03
%S A303393 1,2,4,5,6,7,8,11,15,16,20,22,25,26,28,29,31,33,35,37,40,41,46,50,53,
%T A303393 56,60,61,67,70,71,79,80,83,91,92,96,103,106,110,116,121,125,126,128,
%U A303393 130,131,135,137,140,141,145,146,153,154,158,161,170,172,176
%N A303393 Numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.
%C A303393 The author's conjecture in A303389 has the following equivalent version: Each integer n > 1 can be expressed as the sum of two terms of the current sequence.
%C A303393 This has been verified for all n = 2..2*10^8.
%H A303393 Zhi-Wei Sun, <a href="/A303393/b303393.txt">Table of n, a(n) for n = 1..10000</a>
%H A303393 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 A303393 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 A303393 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 A303393 a(1) = 1 with 1 = 0*(0+1)/2 + 5^0.
%e A303393 a(2) = 2 with 2 = 1*(1+1)/2 + 5^0.
%e A303393 a(3) = 4 with 4 = 2*(2+1)/2 + 5^0.
%t A303393 TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
%t A303393 tab={};Do[Do[If[TQ[m-5^k],tab=Append[tab,m];Goto[aa]],{k,0,Log[5,m]}];Label[aa],{m,1,176}];Print[tab]
%Y A303393 Cf. A000217, A000351, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303389.
%K A303393 nonn
%O A303393 1,2
%A A303393 _Zhi-Wei Sun_, Apr 23 2018