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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

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Last modified July 23 22:37 EDT 2019. Contains 325278 sequences. (Running on oeis4.)