A255160 Least positive integer m with A254885(m) = n.

1, 3, 10, 11, 19, 35, 55, 46, 71, 136, 86, 131, 200, 170, 221, 275, 271, 235, 236, 401, 341, 326, 491, 478, 586, 431, 731, 716, 536, 635, 775, 851, 821, 695, 1040, 950, 1241, 1171, 1160, 1031, 1306, 1115, 1801, 1460, 1706, 1391, 1531, 1685, 1790, 1670, 2081, 1745, 2161, 2021, 1976, 2330, 2350, 2216, 2645, 2615

Offset: 1

Author

Zhi-Wei Sun, Feb 15 2015

Keywords

• nonn

Comments

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

Examples

			 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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• Zhi-Wei Sun, 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), arXiv:1502.03056 [math.NT], 2015.

Crossrefs

Cf. A000217, A000290, A254885.

Programs

Mathematica

TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
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],
Goto[aa]]];Label[bb];Continue,{n,1,60}]