

A188896


Numbers n such that there is no square ngonal number greater than 1.


10



10, 20, 52, 164, 340, 580, 884, 1252, 1684, 2180, 2740, 4052, 4804, 5620, 6500, 7444, 8452, 9524, 10660, 11860, 13124, 14452, 15844, 17300, 18820, 20404, 22052, 25540, 27380, 29284, 31252, 33284, 35380, 37540, 39764, 42052, 44404, 46820, 49300, 51844
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OFFSET

1,1


COMMENTS

It is easy to find squares that are triangular, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no square 10gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (n2)*y^2  (n4)*y has no integer solutions x>1 and y>1.
Chu shows how to transform the equation into a generalized Pell equation. When n has the form 2k^2+2 (A005893), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.
The general case is in A188950.


LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..244
Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781802.


MATHEMATICA

P[n_, k_]:=1/2n(n(k2)+4k); data1=2#^2+2&/@Range[2, 161]; data2=Head[Reduce[m^2==P[n, #] && 1<m && 1<n && !m==n, {m, n}, Integers]]&/@data1; data3=Flatten[Position[data2, Symbol]]; data1[[#]]&/@data3 (* Ant King, Mar 01 2012 *)


CROSSREFS

Cf. A001107 (10gonal numbers), A051872 (20gonal numbers), A188892, A100252, A188950, A005893.
Subsequence of A271624.  Muniru A Asiru, Oct 16 2016
Sequence in context: A115045 A205879 A254030 * A067192 A030004 A271512
Adjacent sequences: A188893 A188894 A188895 * A188897 A188898 A188899


KEYWORD

nonn


AUTHOR

T. D. Noe, Apr 13 2011


STATUS

approved



