All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 16 Next

Displaying 301 – 320 of 341

Showing per page

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

For k = 1,2,... let H k denote the harmonic number ∑ j = 1 k 1 / j . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have ∑ k = 1 p - 1 ( H k ) / ( k 2 k ) ≡ 7 / 24 p B p - 3 ( m o d p ² ) , ∑ k = 1 p - 1 ( H k , 2 ) / ( k 2 k ) ≡ - 3 / 8 B p - 3 ( m o d p ) , and ∑ k = 1 p - 1 ( H ² k , 2 n ) / ( k 2 n ) ≡ ( 6 n + 1 2 n - 1 + n ) / ( 6 n + 1 ) p B p - 1 - 6 n ( m o d p ² ) for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and H k , m : = ∑ j = 1 k 1 / ( j m ) .

Ascents of size less than d in compositions

Maisoon Falah, Toufik Mansour (2011)

Open Mathematics

A composition of a positive integer n is a finite sequence π1π2...πm of positive integers such that π1+...+πm = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if πi+1 ≥ πi+d (respectively, πi < πi+1 < πi + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable...

Currently displaying 301 – 320 of 341