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A188950 Pairs of numbers (n,k) such that there is no n-gonal k-gonal number greater than 1, sorted by the sum n+k and then n. 3
3, 11, 4, 10, 6, 11, 5, 14, 3, 18, 4, 20, 6, 18, 7, 22, 11, 18, 10, 20, 6, 27, 5, 29, 8, 26, 11, 27, 9, 30, 3, 38, 14, 29, 6, 38, 10, 34, 18, 27, 11, 38, 7, 47, 12, 42, 20, 34, 5, 50, 4, 52, 18, 38, 6, 51, 13, 46, 11, 51, 8, 56, 14, 50, 27, 38, 15, 54, 22, 47 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These are n and k such that the generalized Pell equation (k-2)*x^2 - (k-4)*x = (n-2)*y^2 - (n-4)*y has no solution in integers x>1 and y>1. The paper by Chu shows how to solve these equations. A necessary condition for a pair to be in this sequence is (n-2)(k-2) is a square. These (n,k) pairs indicate where the zeros are in triangle A189216, which gives the least n-gonal k-gonal number greater than 1. For triangular (n=3) and square (n=4) numbers, see A188892 and A188896 for lists of k.

LINKS

Table of n, a(n) for n=1..70.

Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.

Eric W. Weisstein, MathWorld: Polygonal Number

EXAMPLE

The pairs begin (3,11), (4,10), (6,11), (5,14), (3,18), (4,20), (6,18).

MATHEMATICA

maxSum=100; Reap[Do[k=s-n; If[k>n && IntegerQ[Sqrt[(n-2)*(k-2)]] && FindInstance[(k-2)*x^2 - (k-4)*x == (n-2)*y^2 - (n-4)*y && x>1 && y>1, {x, y}, Integers] == {}, Sow[{n, k}]], {s, 7, maxSum}, {n, 3, s-3}]][[2, 1]]

CROSSREFS

Cf. A188892, A188896.

Sequence in context: A306367 A226625 A210610 * A308968 A084466 A084462

Adjacent sequences:  A188947 A188948 A188949 * A188951 A188952 A188953

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, Apr 20 2011

STATUS

approved

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Last modified January 20 19:53 EST 2022. Contains 350472 sequences. (Running on oeis4.)