منابع مشابه
On Consecutive Happy Numbers
Let e > 1 and b > 2 be integers. For a positive integer n = ∑k j=0 aj × b j with 0 6 aj < b, define
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A happy number N is defined by the condition S(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, 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 numb...
متن کاملOn the Density of Happy Numbers
The happy function H : N → N sends a positive integer to the sum of the squares of its digits. A number x is said to be happy if the sequence {Hn(x)}∞ n=1 eventually reaches 1 (here H(x) denotes the nth iteration of H on x). A basic open question regarding happy numbers is what bounds on the density can be proved. This paper uses probabilistic methods to reduce this problem to experimentally fi...
متن کاملOn the Difference between Consecutive Ramsey Numbers
It is shown that the classical Ramsey numbers T( m, ta) satisfy r(m,n) 2: r(m,n1) + 2m3,
متن کاملΝοτε Ον Consecutive Abundant Numbers
A positive integer N is called an abundant number if σ(N) > 2N, where σ (N) denotes the suns of the divisors of N including 1 and N. Abundant numbers have been recently investigated by Behrend, Chowla, Davenport , myself, and others ; it has been proved, for example, that they have a density greater than 0 . I prove now the following THEOREM. We can find two constants c l , c 2 such that , for ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2008
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2007.11.009