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

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, 35, 36, 37, 38, 40, 42, 44, 46, 47, 49, 52, 53, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 70, 71, 74, 77, 79, 80

Offset

1,2

Comments

Conjecture: Any integer n > 1 can be written as the sum of two terms of the current sequence.

This is equivalent to the author's conjecture in A303233.

Links

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

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.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Example

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

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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]

Crossrefs

Cf. A000079, A000217, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233.

Sequence in context: A267304 A347528 A002855 * A175233 A346130 A121229

Adjacent sequences: A303231 A303232 A303233 * A303235 A303236 A303237

Keyword

nonn

Author

Zhi-Wei Sun, Apr 20 2018

Status

approved