SEQUENCES OF CONSECUTIVE FACTORADIC HAPPY NUMBERS

摘要

Given a positive integer n, the factorial base representation of n is given by n =Sigma(k)(i=1) a(i) center dot i!, where a(k) not equal 0 and 0 <= a(i) <= i for all 1 <= i <= k. For e >= 1, we define S-e,S- t Z(>= 0) -> Z(>= 0) by S-e,S-t (0) = 0 and S-e,S-t(n)= Sigma(n)(i=1) a(i)(e), for n not equal 0. For l >= 0, we let S-e,t(l),(n) denote the l-th iteration of S-e,S-t, while S-e,t(0)(n)= n. If p is an element of Z(+) satisfies Se,t(p) = p, then we say that p is an e-power factoradic fixed point of S-e,S-t. Moreover, given x E Z+, if p is an e-power factoradic fixed point and if there exists.e E 7L>0 such that S-e,t(l)(x) = p, then we say that x is an e-power factoradic p-happy number. Note an integer n is said to be an e-power factoradic happy number if it is an e-power factoradic 1-happy number. We prove that all positive integers are 1-power factoradic happy and, for 2 <= e <= 4, we prove the existence of arbitrarily long sequences of e-power factoradic p-happy numbers. A curious result establishes that for any e >= 2, the e-power factoradic fixed points of S-e,S-t that are greater than 1 always appear in sets of consecutive pairs. Our last contribution provides the smallest sequences of m consecutive e-power factoradic happy numbers for 2 < e < 5, for some values of m.