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Numbers of the form x*(3*x-1)/2 + 3^y with x and y nonnegative integers.
24

%I #10 Apr 25 2018 14:02:11

%S 1,2,3,4,6,8,9,10,13,14,15,21,23,25,27,28,31,32,36,38,39,44,49,52,54,

%T 60,62,71,73,78,79,81,82,86,93,95,97,101,103,116,118,119,120,126,132,

%U 144,146,148,151,154,172,173,177,179,185

%N Numbers of the form x*(3*x-1)/2 + 3^y with x and y nonnegative integers.

%C The author's conjecture in A303401 has the following equivalent version: Each integer n > 1 can be written as the sum of two terms of the current sequence.

%C This has been verified for all n = 2..7*10^6.

%H Zhi-Wei Sun, <a href="/A303434/b303434.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://math.scichina.com:8081/sciAe/EN/abstract/abstract517007.shtml">On universal sums of polygonal numbers</a>, Sci. China Math. 58(2015), no. 7, 1367-1396.

%H 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 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.

%e a(1) = 1 with 1 = 0*(3*0-1)/2 + 3^0.

%e a(2) = 2 with 2 = 1*(3*1-1)/2 + 3^0.

%e a(5) = 6 with 6 = 2*(3*2-1)/2 + 3^0.

%e a(6) = 8 with 8 = 2*(3*2-1)/2 + 3^1.

%t PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);

%t tab={};Do[Do[If[PenQ[m-3^k],n=n+1;tab=Append[tab,m];Goto[aa]],{k,0,Log[3,m]}];Label[aa],{m,1,185}];Print[tab]

%Y Cf. A000244, A000326, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Apr 23 2018