Journal of Integer Sequences, Vol. 22 (2019), Article 19.7.6 |

Department of Mathematics

Iowa State University

Carver Hall, 411 Morrill Road

Ames, IA 50011

USA

Grant Fickes

Department of Mathematics

Kutztown University of Pennsylvania

15200 Kutztown Road

Kutztown, PA 19530

USA

Eugene Fiorini

Department of Mathematics

Muhlenberg College

2400 Chew Street

Allentown, PA 18104

USA

Edgar Jaramillo Rodriguez

Department of Mathematics

University of California, Davis

1 Shields Avenue

Davis, CA 95616

USA

Eric Jovinelly

Department of Mathematics

Notre Dame University

255 Hurley

Notre Dame, IN 46556

Tony W. H. Wong

Department of Mathematics

Kutztown University of Pennsylvania

15200 Kutztown Road

Kutztown, PA 19530

USA

**Abstract:**

The Catalan triangle is an infinite lower-triangular matrix that
generalizes the Catalan numbers. The entries of the Catalan triangle,
denoted by *c*_{n,k},
count the number of shortest lattice paths from
(0,0) to (*n*,*k*) that do not go above the main diagonal. This paper
studies the occurrence of primes and perfect powers in the Catalan
triangle. We prove that no prime powers except 2, 5, 9, and 27
appear in the Catalan triangle when *k* ≥ 2. We further prove that
*c*_{n,k} are not perfect semiprime powers when *k* ≥ 3. Finally, by
assuming the *abc* conjecture, we prove that aside from perfect squares
when *k* = 2,
there are at most finitely many perfect powers among
*c*_{n,k}
when *k* ≥ 2.

(Concerned with sequences A275481 A275586 A317027.)

Received January 30 2019;
revised versions received October 18 2019; October 31 2019.
Published in *Journal of Integer Sequences*,
November 8 2019.

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