A277449 Numbers n such that there is exactly one nontrivial square n-gonal number.

34, 74, 100, 130, 202, 244, 290, 394, 452, 514, 650, 724, 802, 970, 1060, 1154, 1354, 1460, 1570, 1802, 1924, 2050, 2314, 2452, 2594, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4234, 4420, 4610, 5002, 5204, 5410, 5834, 6052, 6274, 6730, 6964, 7202, 7690, 7940, 8194, 8714, 8980, 9250, 9802, 10084, 10370, 10954, 11252, 11554, 12170, 12484, 12802, 13450, 13780

Offset: 1

Author

Muniru A Asiru, Oct 16 2016

Keywords

• nonn

Comments

There are infinitely many squares that are triangular, pentagonal, hexagonal, etc. Also there is no square 10-gonal number, 20-gonal number, 52-gonal number, 64-gonal number, etc. greater than 1 (see A188896). Other than the trivial square n-gonal numbers 0 and 1, there is exactly one square 34-gonal number, one square 74-gonal number, one square 100-gonal number, one square 130-gonal number, etc.

Examples

			For n = 34, the square 34-gonal numbers are 0, 1, 196.
For n = 74, the square 74-gonal numbers are 0, 1, 2601.
For n = 100, the square 100-gonal numbers are 0, 1, 100.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..238

Crossrefs

Cf. A005893, A188896, A271624.

Programs

GAP

G:=[];; for g in [5..100000]  do for r in [1..5000] do if 2*g-4=r^2 then Add(G,g); fi; od; od; G; Length(G);
F:=List(G,g->[g,DivisorsInt((g-4)^2)]);;
N:=List([1..Length(F)], i->List([1..Length(F[i][2])],j->[F[i][1],((F[i][1]-4)*(F[i][1]-4+2*F[i][2][j])+F[i][2][j]^2)/((4*F[i][1]-8)*F[i][2][j])] ) );;
N1:=Filtered(List(List([1..Length(N)],k->Filtered(N[k], l->IsPosInt(l[2]))),Set),o->Length(o)>=2);
N2:=Set(Flat(List([1..Length(N1)],i->List([1..Length(N1[i])],j->N1[i][j][1]))));