A254680 Least positive integer m with A254661(m) = n.

1, 3, 7, 17, 21, 51, 66, 72, 157, 147, 121, 136, 246, 297, 332, 367, 402, 506, 547, 577, 677, 796, 892, 731, 926, 1216, 1116, 976, 1181, 1402, 1556, 1416, 1507, 1496, 2287, 1622, 1977, 2112, 1942, 2131, 2017, 2882, 2767, 2501, 3162, 3671, 3097, 3187, 3047, 3762, 3867, 2952, 4356, 4111, 4826, 5112, 5211, 4811, 4686, 5461

Offset: 1

Author

Zhi-Wei Sun, Feb 05 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. Moreover, no term is divisible by 5, and no term with n>2 is congruent to 3 modulo 5.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..170
• Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Crossrefs

Cf. A000217, A000290, A005449, A160325, A254661, A254677.

Programs

Maple

a(3) = 7 since 7 is the first positive integer that can be written as x*(x+1)/2 + (2y)^2 + z*(3*z+1)/2 (with x,y,z nonnegative integers) in exactly 3 ways. In fact, 7 = 0*1/2 + 0^2 +2*(3*2+1)/2 = 1*2/2 + 2^2 +1*(3*1+1)/2 = 2*3/2 + 2^2 + 0*(3*0+1)/2.

Mathematica

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
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}];
If[r==n,Print[n," ",m];Goto[bb],Goto[aa]]]; Label[bb];Continue,{n,1,60}]