A254677 - OEIS

%I #13 Feb 27 2018 03:32:36

%S 2,1,5,19,15,22,37,92,71,156,136,222,206,211,257,292,506,402,577,521,

%T 632,789,682,796,742,1006,1046,1192,1346,1482,1312,1507,2021,1522,

%U 2172,1977,1962,2007,2161,2479,2502,3047,2761,2326,3097,2876,3316,3216,3421,3386,3902,3652,4406,4356,4587,4492,4342,4917,4811,5472

%N Least positive integer m with A160325(m) = n.

%C Conjecture: a(n) exists for any n > 0. Moreover, no term a(n) is congruent to 3 modulo 5.

%H Zhi-Wei Sun, <a href="/A254677/b254677.txt">Table of n, a(n) for n = 1..200</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:0905.0635 [math.NT], 2009-2015.

%e a(3) = 5 since 5 is the least positive integer that can be written as x(x+1)/2 + (2y)^2 + z(3z-1)/2 (with x,y,z nonnegative integers) in exactly 3 ways. In fact, 5 = 0*1/2 + 0^2 + 2*(3*2-1)/2 = 0*1/2 + 2^2 + 1*(3*1-1)/2 = 1*2/2 + 2^2 + 0*(3*0-1)/2.

%t TQ[n_]:=IntegerQ[Sqrt[8n+1]]

%t Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-4y^2-z(3z-1)/2],r=r+1;If[r>n,Goto[aa]]],{y,0,Sqrt[m/4]},{z,0,(Sqrt[24(m-4y^2)+1]+1)/6}];

%t If[r==n,Print[n," ", m];Goto[bb],Goto[aa]]];Label[bb];Continue,{n,1,60}]

%Y Cf. A000217, A000290, A000326, A160325.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Feb 05 2015