A280472 Number of ways to write n as the sum of an octagonal number (A000567), a second octagonal number (A045944), and a strict partition number (A000009).

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

Offset: 1

Author

Zhi-Wei Sun, Jan 04 2017

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) lim_n a(n)/(log n)^2 = 1/Pi^2.

On 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.

See also A280455 for a similar conjecture of the author involving the partition function.

Examples

			a(1) = 1 since 1 = 0*(3*0-2) + 0*(3*0+2) + A000009(2).
a(50) = 2 since 50 = 4*(3*4-2) + 1*(3*1+2) + A000009(7) = 4*(3*4-2) + 0*(3*0+2) + A000009(10).
a(1399) = 1 since 1399 = 1*(3*1-2) + 18*(3*18+2) + A000009(32).
		

Links

• 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.
• Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.

Crossrefs

Cf. A000009, A000567, A045944, A280386, A280455.

Programs

Mathematica

Oct[n_]:=Oct[n]=IntegerQ[Sqrt[3n+1]]&&Mod[Sqrt[3n+1],3]==1;
q[n_]:=q[n]=PartitionsQ[n];
Do[r=0;m=2;Label[bb];If[q[m]>n,Goto[cc]];Do[If[Oct[n-q[m]-x(3x-2)],r=r+1],{x,0,(Sqrt[3(n-q[m])+1]+1)/3}];m=m+If[m<3,2,1];Goto[bb];Label[cc];Print[n," ",r];Continue,{n,1,80}]