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, 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

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 * A357115 A308968 A084466

Adjacent sequences: A188947 A188948 A188949 * A188951 A188952 A188953

Keyword

nonn,tabf

Author

T. D. Noe, Apr 20 2011

Status

approved