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%I #19 Jan 04 2017 03:23:04
%S 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,
%T 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,
%U 8,7,4,6,7,9,3,5,4,8,7,9,13
%N 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.
%C Conjecture: (i) a(n) > 0 for all n > 0.
%C (ii) lim_n a(n)/(log n)^2 = 1/Pi^2.
%C 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.
%C 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.
%C As the main term of log p(n) is Pi*sqrt(2n/3), the partition function p(n) eventually grows faster than any polynomial.
%C See also A280472 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A280455/b280455.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/160p.pdf">Problems on combinatorial properties of primes</a>, 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.
%e a(1) = 1 since 1 = 0*(3*0-1)/2 + 0*(3*0+1)/2 + p(1).
%e 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).
%e a(2771) = 1 since 2771 = 35*(3*35-1)/2 + 25*(3*25+1)/2 + p(1).
%e a(9426) = 1 since 9426 = 4*(3*4-1)/2 + 79*(3*79+1)/2 + p(3).
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t p[n_]:=p[n]=PartitionsP[n];
%t Pen[n_]:=Pen[n]=SQ[24n+1]&&Mod[Sqrt[24n+1],6]==1;
%t 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}]
%Y Cf. A000041, A000217, A000326, A005449, A280386, A280444, A280472.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jan 03 2017
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