Results 31 to 40 of about 37,250 (282)

Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion

Mathematics, 2022
Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers.
Kwang-Wu Chen
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Congruences for harmonic sums [PDF]

Notes on Number Theory and Discrete Mathematics, 2023
Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p². In this paper, we improve the harmonic sums Hₚ(n)=Σ_{l₁+l₂+...+lₙ=p, l₁,l₂,...,lₙ>0} 1/l₁+l₂+...+lₙ to supercongruences ...
Yining Yang, Peng Yang
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Harmonic numbers and finite groups [PDF]

Rendiconti del Seminario Matematico della Università di Padova, 2014
Given a finite group G , let {\tau} (G) be the number of normal subgroups of G ...
Baishya, Sekhar Jyoti, Das, Ashish Kumar
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Some Identities with Special Numbers

Cumhuriyet Science Journal, 2022
In this paper, we derive new identities which are related to some special numbers and generalized harmonic numbers H_n (α) by using the argument of the generating function given in [3] and comparing the coefficients of the generating functions.
Sibel Koparal, Neşe Ömür, Kübra Nur Südemen   +2 more
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On q-generalized hyperharmonic numbers with two parameters [PDF]

Notes on Number Theory and Discrete Mathematics
In this paper, we introduce q-generalized harmonic numbers with two parameters ω and ξ, H_{μ,λ,q}(ω,ξ) for integers μ, λ such that λ ≥ μ. With the help of these numbers, we define a new family of numbers which is called q-generalized hyperharmonic ...
Neşe Ömür, Sibel Koparal, Ömer Duran
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Odd harmonic numbers exceed 10²⁴ [PDF]

Mathematics of Computation, 2010
A numbern>1n>1is harmonic ifσ(n)∣nτ(n)\sigma (n)\mid n\tau (n), whereτ(n)\tau (n)andσ(n)\sigma (n)are the number of positive divisors ofnnand their sum, respectively. It is known that there are no odd harmonic numbers up to101510^{15}. We show here that, for any odd numbern>106n>10^6,τ(n)≤n1/3\tau (n)\le n^{1/3}.
Cohen, Graeme L., Sorli, Ronald M.
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Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers [PDF]

Notes on Number Theory and Discrete Mathematics, 2023
We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove Σⁿₖ₌ₒ((-1)ᵏhₖ/4ᵏ)$binom{2k}{k}$Gₙ₋ₖ = ((-1)ⁿ⁻¹/2^²ⁿ⁻¹)
Necdet Batır, Anthony Sofo
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Some identities involving harmonic numbers [PDF]

Mathematics of Computation, 1990
Let H n {H_n} denote the nth harmonic number. Explicit formulas for sums of the form ∑ a k H k \sum {a_k}{H_k} or ∑ a
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Some results on q-harmonic number sums

Advances in Difference Equations, 2018
In this paper, we establish some relations involving q-Euler type sums, q-harmonic numbers and q-polylogarithms. Then, using the relations obtained with the help of q-analog of partial fraction decomposition formula, we develop new closed form ...
Xin Si
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Series of Convergence Rate −1/4 Containing Harmonic Numbers

Axioms, 2023
Two general transformations for hypergeometric series are examined by means of the coefficient extraction method. Several interesting closed formulae are shown for infinite series containing harmonic numbers and binomial/multinomial coefficients.
Chunli Li, Wenchang Chu
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