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A188896 Numbers n such that there is no square n-gonal number greater than 1. 12
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
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
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 13 2011
STATUS
approved

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Last modified March 19 03:21 EDT 2024. Contains 370952 sequences. (Running on oeis4.)