The On-Line Encyclopedia of Integer Sequences (OEIS)

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A303428

Number of ways to write n as x*(3*x-2) + y*(3*y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.
(history; published version)

#14 by Bruno Berselli at Tue Apr 24 05:52:00 EDT 2018

STATUS

proposed

approved

#13 by Zhi-Wei Sun at Mon Apr 23 21:34:36 EDT 2018

STATUS

editing

proposed

#12 by Zhi-Wei Sun at Mon Apr 23 21:34:01 EDT 2018

COMMENTS

a(n) > 0 for all n = 2..23*10^8. Those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082). 76683391 is the least integer n > 1 not representable as the sum of two generalized octagonal numbers and two powers of 2.

See also A303363 and A303389 , A303401 and A303432 for similar conjectures.

CROSSREFS

Cf. A000244, A000567, A001082, A045944, A280472, A303233, A303338, A303363, A303389, A303393, A303399, A303401, A303432.

STATUS

proposed

editing

#11 by Zhi-Wei Sun at Mon Apr 23 18:17:00 EDT 2018

STATUS

editing

proposed

#10 by Zhi-Wei Sun at Mon Apr 23 18:16:35 EDT 2018

COMMENTS

a(n) > 0 for all n = 2..2*10^8. Note that those Those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082). 76683391 is the least integer n > 1 not representable as the sum of two generalized octagonal numbers and two powers of 2.

EXAMPLE

a(4360) = 4 with 4360 = (-35)*(3*(-35)-2) + (-13)*(3*(-13)-2) + 3^0 + 3^4 = (-37)*(3*(-37)-2) + (-7)*(3*(-7)-2) + 3^2 + 3^2 = (-27)*(3*(-27)-2) + (-23)*(3*(-23)-2) + 3^5 + 3^5 = (-25)*(3*(-25)-2) + (-1)*(3*(-1)-2) + 3^5 + 3^7.

STATUS

proposed

editing

#9 by Zhi-Wei Sun at Mon Apr 23 17:45:45 EDT 2018

STATUS

editing

proposed

#8 by Zhi-Wei Sun at Mon Apr 23 17:45:21 EDT 2018

COMMENTS

a(n) > 0 for all n = 2..2*10^8. Note that those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082).

STATUS

proposed

editing

#7 by Zhi-Wei Sun at Mon Apr 23 13:24:26 EDT 2018

STATUS

editing

proposed

#6 by Zhi-Wei Sun at Mon Apr 23 13:22:26 EDT 2018

COMMENTS

See also A303363 and A303389 for similar conjectures.

EXAMPLE

a(4) = 2 with 4 = 1*(3*1-2) + 1*(3*1-2) + 3^0 + 3^0 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^1.

#5 by Zhi-Wei Sun at Mon Apr 23 13:17:20 EDT 2018

LINKS

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.

Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.

CROSSREFS

Cf. A000244, A000567, A001082, A045944, A280472, A303233, A303338, A303363, A303389, A303393, A303399.