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A188896 Numbers n such that there is no square n-gonal 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 (list; graph; refs; listen; history; text; internal format)
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 10-gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (n-2)*y^2 - (n-4)*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), 781-802.

MATHEMATICA

P[n_, k_]:=1/2n(n(k-2)+4-k); 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 (10-gonal numbers), A051872 (20-gonal 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

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Last modified November 21 03:01 EST 2018. Contains 317427 sequences. (Running on oeis4.)