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The only known odd Catalan numbers that do not have last digit 5 are C 0 = 1, C 1 = 1, C 7 = 429, C 31, C 127 and C 255. The odd Catalan numbers, C n for n = 2 k − 1, do not have last digit 5 if n + 1 has a base 5 representation containing 0, 1 and 2 only, except in the least significant place, which could also be a 3. [3]
The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 (sequence A163209 in the OEIS) with the latter two being squares of Wieferich primes. In general, if p is a Wieferich prime, then p 2 is a Catalan pseudoprime.
In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form + according to Hardy and Littlewood's Conjecture F. However, it is an unsolved problem (one of Landau's problems ) whether there are even infinitely many primes of this form.
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]
There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other. [2] The crossing pattern of chords in a chord diagram may be described by a circle graph , the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross.
The layoffs concerned a number of Republican lawmakers, who privately warned the Trump administration that such cuts could hamper the government’s bird flu response and asked them to reconsider ...
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
[1] Lobb numbers form a natural generalization of the Catalan numbers, which count the complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L 0,n. [2] They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the n th Catalan number. [3]
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