A280472 -id:A280472 - cp's OEIS frontend

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

0, 1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 5, 2, 3, 3, 2, 2, 5, 4, 5, 2, 3, 5, 2, 3, 5, 4, 7, 2, 4, 5, 3, 4, 6, 4, 7, 3, 6, 6, 4, 4, 5, 5, 9, 5, 6, 6, 2, 5, 5, 7, 8, 4, 5, 4, 4, 4, 6, 6, 8, 3, 6, 6, 3, 4, 6, 7, 8, 5, 8, 6, 5, 4, 6, 7, 8, 6, 6, 6, 2

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. Moreover, any integer n > 1 can be written as x*(3*x+2) + y*(3*y+2) + 3^z + 3^w, where x is an integer and y,z,w are nonnegative integers.
a(n) > 0 for all n = 2..3*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 A303389, A303401 and A303432 for similar conjectures.

			a(2) = 1 with 2 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^0.
a(3) = 1 with 3 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^0.
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.
a(5) = 1 with 5 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^1.
a(9) = 1 with 9 = (-1)*(3*(-1)-2) + 0*(3*0-2) + 3^0 + 3^1.
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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
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.

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

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[3(n-3^j-3^k)+2],Do[If[SQ[3(n-3^j-3^k-x(3x-2))+1],r=r+1],{x,-Floor[(Sqrt[3(n-3^j-3^k)/2+1]-1)/3],(Sqrt[3(n-3^j-3^k)/2+1]+1)/3}]],
{j,0,Log[3,n/2]},{k,j,Log[3,n-3^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A280455 Number of ways to write n as x(3x-1)/2 + y(3y+1)/2 + p(z), where x,y,z are nonnegative integers with z > 0, and p(.) is the partition function given by A000041.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 3, 5, 4, 5, 2, 4, 4, 5, 5, 4, 6, 5, 5, 3, 4, 6, 7, 3, 5, 3, 8, 3, 8, 7, 6, 5, 4, 7, 3, 4, 6, 8, 4, 5, 4, 12, 5, 8, 5, 6, 4, 5, 8, 5, 4, 7, 7, 6, 5, 7, 8, 5, 9, 6, 6, 5, 10, 8, 6, 3, 7, 8, 7, 4, 6, 7, 9, 3, 5, 4, 8, 7, 9, 13

Offset: 1

Zhi-Wei Sun, Jan 03 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) lim_n a(n)/(log n)^2 = 1/Pi^2.
This is similar to the author's conjecture in A280386. At the author's request, Prof. Qing-Hu Hou at Tianjin Univ. has verified part (i) of the above conjecture for n up to 10^9.
We also have some other similar conjectures. For example, we conjecture that any positive integer can be expressed as the sum of two triangular numbers and a partition number.
As the main term of log p(n) is Pi*sqrt(2n/3), the partition function p(n) eventually grows faster than any polynomial.
See also A280472 for a similar conjecture.

			a(1) = 1 since 1 = 0*(3*0-1)/2 + 0*(3*0+1)/2 + p(1).
a(2) = 2 since 2 = 1*(3*1-1)/2 + 0*(3*0+1)/2 + p(1) = 0*(3*0-1)/2 + 0*(3*0+1)/2 + p(2).
a(2771) = 1 since 2771 = 35*(3*35-1)/2 + 25*(3*25+1)/2 + p(1).
a(9426) = 1 since 9426 = 4*(3*4-1)/2 + 79*(3*79+1)/2 + p(3).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Cf. A000041, A000217, A000326, A005449, A280386, A280444, A280472.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
p[n_]:=p[n]=PartitionsP[n];
Pen[n_]:=Pen[n]=SQ[24n+1]&&Mod[Sqrt[24n+1],6]==1;
Do[r=0;m=1;Label[bb];If[p[m]>n,Goto[cc]];Do[If[Pen[n-p[m]-x(3x-1)/2],r=r+1],{x,0,(Sqrt[24(n-p[m])+1]+1)/6}];m=m+1;Goto[bb];Label[cc];Print[n," ",r];Label[aa];Continue,{n,1,80}]

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A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

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A280455 Number of ways to write n as x(3x-1)/2 + y(3y+1)/2 + p(z), where x,y,z are nonnegative integers with z > 0, and p(.) is the partition function given by A000041.

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cp's OEIS Frontend

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.

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A280455 Number of ways to write n as x*(3x-1)/2 + y*(3y+1)/2 + p(z), where x,y,z are nonnegative integers with z > 0, and p(.) is the partition function given by A000041.

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Mathematica

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

A280455 Number of ways to write n as x(3x-1)/2 + y(3y+1)/2 + p(z), where x,y,z are nonnegative integers with z > 0, and p(.) is the partition function given by A000041.