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On Consecutive Happy Numbers
 Journal of Number Theory
"... Abstract. Let e � 1 and b � 2 be integers. For a positive integer n = ∑k j=0 aj × bj with 0 � aj < b, define k∑ Te,b(n) = a e j. n is called (e, b)happy if T r e,b (n) = 1 for some r � 0, where T r e,b is the rth iteration of Te,b. In this paper, we prove that there exists arbitrarily long se ..."
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Abstract. Let e � 1 and b � 2 be integers. For a positive integer n = ∑k j=0 aj × bj with 0 � aj < b, define k∑ Te,b(n) = a e j. n is called (e, b)happy if T r e,b (n) = 1 for some r � 0, where T r e,b is the rth iteration of Te,b. In this paper, we prove that there exists arbitrarily long sequences of consecutive (e, b)happy numbers provided that e − 1 is not divisible by p − 1 for any prime divisor p of b − 1. j=0 1.
Sequences of Generalized Happy Numbers with Small Bases
"... For bases b ≤ 5 and exponents e ≥ 2, there exist arbitrarily long finite sequences of dconsecutive epower bhappy numbers for a specific d = d(e, b), which is shown to be minimal possible. 1 ..."
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For bases b ≤ 5 and exponents e ≥ 2, there exist arbitrarily long finite sequences of dconsecutive epower bhappy numbers for a specific d = d(e, b), which is shown to be minimal possible. 1
Article 10.4.8 Semihappy Numbers
"... We generalize the concept of happy number as follows. Let e = (e0, e1,....) be a sequence with e0 = 2 and ei = {1, 2} for i> 0. Define Se: Z + → Z + by n∑ Se ai10 i n∑ = a ei i. i=0 If S k e(a) = 1 for some k ∈ Z +, then we say that a is a semihappy number or, more precisely, an esemihappy numb ..."
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We generalize the concept of happy number as follows. Let e = (e0, e1,....) be a sequence with e0 = 2 and ei = {1, 2} for i> 0. Define Se: Z + → Z + by n∑ Se ai10 i n∑ = a ei i. i=0 If S k e(a) = 1 for some k ∈ Z +, then we say that a is a semihappy number or, more precisely, an esemihappy number. In this paper, we determine fixed points and cycles of the functions Se and discuss heights of semihappy numbers. We also prove that for each choice of e, there exist arbitrarily long finite sequences of consecutive esemihappy numbers. i=0 1
Smallest examples of strings of consecutive happy numbers
"... Abstract A happy number N is defined by the condition S n (N ) = 1 for some number n of iterations of the function S, where S(N ) is the sum of the squares of the digits of N . Up to 10 20 , the longest known string of consecutive happy numbers was length five. We find the smallest string of consec ..."
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Abstract A happy number N is defined by the condition S n (N ) = 1 for some number n of iterations of the function S, where S(N ) is the sum of the squares of the digits of N . Up to 10 20 , the longest known string of consecutive happy numbers was length five. We find the smallest string of consecutive happy numbers of length 6, 7, 8, . . . , 13. For instance, the smallest string of six consecutive happy numbers begins with N = 7899999999999959999999996. We also find the smallest sequence of 3consecutive cubic happy numbers of lengths 4, 5, 6, 7, 8, and 9.