A254631 -id:A254631 - cp's OEIS frontend

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

Zhi-Wei Sun, Feb 04 2015

nonn

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.

			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).

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.

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

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}]

A253187 Number of ordered ways to write n as the sum of a pentagonal number, a second pentagonal number and a generalized decagonal number.

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 1, 3, 4, 2, 2, 1, 4, 3, 3, 4, 2, 3, 1, 3, 2, 2, 5, 3, 3, 3, 3, 6, 3, 6, 4, 2, 3, 1, 7, 2, 4, 5, 5, 4, 1, 5, 5, 2, 3, 4, 4, 5, 5, 5, 3, 5, 7, 6, 4, 3, 1, 6, 6, 8, 5, 3, 6, 4, 7, 4, 2, 6, 5, 5, 3, 4, 8, 3, 3, 3, 6, 6, 7, 9, 6, 2, 5, 6, 7, 7, 4, 6, 6, 7, 5, 3, 10, 6, 3, 4, 5, 7, 3, 10, 7

Offset: 0

Zhi-Wei Sun, Apr 07 2015

nonn

Conjecture: a(n) > 0 for all n. Also, for any ordered pair (k,m) among (5,7), (5,9), (5,13), (6,5), (6,7), (7,5), each nonnegative integer n can be written as the sum of a k-gonal number, a second k-gonal number and a generalized m-gonal number.

See also the author's similar conjectures in A254574, A254631, A255916 and the two linked papers.

			a(33) = 1 since 33 = 0*(3*0-1)/2 + 4*(3*4+1)/2 + 1*(4*1+3).
a(56) = 1 since 56 = 4*(3*4-1)/2 + 2*(3*2+1)/2 + 3*(4*3+3).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
Zhi-Wei Sun, On 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.

Cf. A000326, A000384, A000566, A005449, A014105, A074377, A085787, A147875, A254574, A254631.

DQ[n_]:=IntegerQ[Sqrt[16n+9]]
Do[r=0;Do[If[DQ[n-x(3x-1)/2-y(3y+1)/2],r=r+1],{x,0,(Sqrt[24n+1]+1)/6},{y,0,(Sqrt[24(n-x(3x-1)/2)+1]-1)/6}];
Print[n," ",r];Continue,{n,0,100}]

cp's OEIS Frontend

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

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