A303434 Numbers of the form x*(3*x-1)/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

Offset

1,2

Comments

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..7*10^6.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Example

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.

Mathematica

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]

Crossrefs

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

Sequence in context: A349607 A060306 A158614 * A362399 A080823 A117925

Adjacent sequences: A303431 A303432 A303433 * A303435 A303436 A303437

Keyword

nonn

Author

Zhi-Wei Sun, Apr 23 2018

Status

approved