To give examples of large combinatorial problems D. Knuth modified W. Ackermann's example of a recursive, but not primitive recursive, function to produce a class of nonassociative compositions. These arrow compositions, which we call krata, are defined on the positive integers by settingB1T=BTBD1=BBD+1(T+1)=BD(BD+1T).The function k(B, D, T) = BDT, which usually takes on large values, has interesting periodicity properties modulo every positive integer M. For fixed B, D, T 2 the sequences {BDn}, {BnT} and {Bnn} are eventually constant modulo M. Also {nDT}, {nDn}, {nnT} and {nnn} are eventually periodic modulo M. An algorithm for calculating BDT modulo M is given. 1983.
