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Acta Mathematica Academiae Scientia,um Hungaricae Tomus 33 (1--2), (1979), pp. 71--80. SOME UNCONVENTIONAL PROBLEMS IN NUMBER THEORY By P. ERDOS (Budapest), member of the Academy Dedicated to the 80th birthday of my friend George Alexits In the paper, we will mostly deal with arithmetic functions, primes, divisors, sieve processes and consecutive integers. 1. Letfbe an arithmetic function. The integer n is called a barrier forfif (1) m + f(m) < n for every m < n. Perhaps I should explain why I considered (1). In the early 1950's, van Wijn- gaarden told me the following conjecture. Put al(n) = a(n), the sum of divisors of n, and ak(n) = al(a~_i(n)). Is it true that there is essentially only one sequence a~(n) (k = 1, 2, 3 .... ) ? In other words, if m and n are distinct integers, are there integers k and l such that ak(m ) = al(n ) ? Such a conjecture is usually hopeless to prove Or disprove. Selfridge and others made some computer experiments and believe that the conjecture is false. I tried to find an airthmetic function for which an analogous conjecture is true and can be proved. Putf,(n) = n +
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: Jun 21, 2005
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