Handbook of Number Theory II (v. 2) by Jozsef Sandor, Borislav Crstici

By Jozsef Sandor, Borislav Crstici

This instruction manual makes a speciality of a few vital issues from quantity concept and Discrete arithmetic. those contain the sum of divisors functionality with the numerous previous and new matters on ideal numbers; Euler's totient and its many points; the Möbius functionality besides its generalizations, extensions, and purposes; the mathematics services concerning the divisors or the digits of a host; the Stirling, Bell, Bernoulli, Euler and Eulerian numbers, with connections to numerous fields of natural or utilized arithmetic. every one bankruptcy is a survey and will be considered as an encyclopedia of the thought of box, underlining the interconnections of quantity concept with Combinatorics, Numerical arithmetic, Algebra, or likelihood Theory.

This reference paintings could be priceless to experts in quantity conception and discrete arithmetic in addition to mathematicians or scientists who desire entry to a few of those leads to different fields of research.

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Hornfeck [146] who showed that for V (x) = car d{n ≤ x : n perfect} (56) one has V (x) < x 1/2 (57) A year later [147] he showed that V (x) 1 lim sup √ ≤ √ x x→∞ 2 5 30 (58) PERFECT NUMBERS Kanold [169] improved these to V (x) < c x 1/4 log x log log x (59) while Hornfeck and E. Wirsing [148], resp. Wirsing [328] proved that V (x) < exp c log x log log log x log log x (60) c log x log log x (61) and V (x) < exp It follows from this result that even if some odd perfect numbers do exist, at least they are less numerous than, for example, the primes.

Pedersen [231] found 139 new infinitary (43) perfect numbers. Cohen proves also that the only infinitary perfect numbers not divisible by 8 are 6, 60 and 90. (44) We note that D. Suryanarayana [306] and K. Alladi [6] have given different generalizations from above for unitary divisors, thus obtaining other notions of k-ary divisors. e. if the unitary harmonic mean H ∗ (n) = nd ∗ (n)/σ ∗ (n) is an integer. The main properties of unitary harmonic numbers were studied by Hagis and Lord [135], but it was earlier introduced also by N.

J. S´andor [270] (see pp. e. such that σ ∗ (σ ∗ (n)) = 2n − 1 (22) are n = 1 and n = 3. Sitaramaiah and Subbarao [290] rediscovered this result. They study more profoundly the unitary superperfect numbers n with σ ∗ (σ ∗ (n)) = 2n (23) The first ten such numbers are 2, 9, 165, 238, 1640, 4320, 10250, 10824, 13500 and 23760. There are not known odd unitary superperfect numbers other than 9 and 165, and Sitaramaiah and Subbarao conjecture that the set of such numbers is finite. (24) 47 CHAPTER 1 In fact 165 is the single such number, having three distinct prime factors.

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