A254639 Least positive integer m such that A254631(m) = n.

2, 1, 6, 16, 27, 62, 71, 92, 122, 161, 176, 216, 286, 386, 351, 491, 577, 492, 781, 866, 1023, 617, 736, 1002, 1504, 1441, 1402, 1297, 1451, 1562, 1842, 2166, 1682, 1331, 2626, 2311, 2332, 2969, 3177, 2761, 2876, 3641, 3261, 3697, 3586, 4894, 3576, 3921, 4482, 4542

Offset: 1

Author

Zhi-Wei Sun, Feb 04 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. Moreover, no term a(n) is divisible by 5.

It seems that no term a(n) is congruent to 8 modulo 10.

Examples

			a(3) = 6 since 6 is the least positive integer m with A254631(m) = 3. Note that 6 = 0*1/2 + 1*(3*1+2) + 1*(3*1-2) = 1*2/2 + 1*(3*1+2) + 0*(3*0-2) = 3*4/2 + 0*(3*0+2) + 0*(3*0-2).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..160
• Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Crossrefs

Cf. A000217, A000567, A045944, A254573, A254574, A254595, A254617, A254631.

Programs

Mathematica

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-y(3y+2)-z(3z-2)],r=r+1;If[r>n, Goto[aa]]],{y,0,(Sqrt[3m+1]-1)/3},{z,0,(Sqrt[3(m-y(3y+2))+1]+1)/3}];
If[r==n,Print[n, " ", m];Goto[bb],Goto[aa]]];Label[bb];Continue,{n,1,50}]