proposed

approved

editing

proposed

This has been verified for all n = 2..37*10^6.

approved

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proposed

approved

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proposed

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

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.

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.

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.

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

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

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.

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

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

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

allocated for Zhi Numbers of the form x*(3*x-Wei Sun1)/2 + 3^y with x and y nonnegative integers.

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, 60, 62, 71, 73, 78, 79, 81, 82, 86, 93, 95, 97, 101, 103, 116, 118, 119, 120, 126, 132, 144, 146, 148, 151, 154, 172, 173, 177, 179, 185

1,2

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.

This has been verified for all n = 2..3*10^6.

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

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

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

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

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

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]

allocated

nonn

Zhi-Wei Sun, Apr 23 2018

approved

editing

allocated for Zhi-Wei Sun

allocated

approved