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

1,1

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.

Links

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

Example

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.

Prog

(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]))));

Crossrefs

Cf. A005893, A188896, A271624.

Sequence in context: A079863 A154095 A184066 * A063533 A216432 A118619

Adjacent sequences: A277446 A277447 A277448 * A277450 A277451 A277452

Keyword

nonn

Author

Muniru A Asiru, Oct 16 2016

Status

approved