On quotients of values of Euler's function on the Catalan numbers

On the Catalan Numbers and Some of Their Identities

“…These numbers occupy a pivotal position in combinatorial mathematics, as many counting problems are closely related to Catalan numbers, and some famous examples can be found in R. P. Stanley [1]. Many papers related to the Catalan numbers and other special sequences can also be found in references [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], especially the works of T. Kim et al give a series of new identities for the Catalan numbers, see [9][10][11][12][13][14], these are important results in the related field.…”