A255160 - OEIS

%I #6 Feb 15 2015 15:59:03

%S 1,3,10,11,19,35,55,46,71,136,86,131,200,170,221,275,271,235,236,401,

%T 341,326,491,478,586,431,731,716,536,635,775,851,821,695,1040,950,

%U 1241,1171,1160,1031,1306,1115,1801,1460,1706,1391,1531,1685,1790,1670,2081,1745,2161,2021,1976,2330,2350,2216,2645,2615

%N Least positive integer m with A254885(m) = n.

%C Conjecture: a(n) exists for any n > 0. Moreover, no term a(n) is congruent to 2 or 4 or 7 modulo 10.

%H Zhi-Wei Sun, <a href="/A255160/b255160.txt">Table of n, a(n) for n = 1..200</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">Universal sums a*x^2+b*y^2+f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+f(z)</a>, arXiv:1502.03056 [math.NT], 2015.

%e  a(3) = 10 since 10 is the least positive integer which can be written as the sum of two squares and a positive triangular number in exactly 3 ways. In fact, 10 = 0^2 + 0^2 + 4*5/2 = 0^2 + 2^2 + 3*4/2 = 0^2 + 3^2 + 1*2/2.

%t TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]

%t Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-x^2-y^2],r=r+1;If[r>n,Goto[aa]]],{x,0,Sqrt[m/2]},{y,x,Sqrt[m-x^2]}];If[r==n,Print[n," ",m];Goto[bb],

%t Goto[aa]]];Label[bb];Continue,{n,1,60}]

%Y Cf. A000217, A000290, A254885.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Feb 15 2015