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1
Content available remote

Acknowledgment of priority: "On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ" (Colloq. Math. 92 (2002), 111-130)

100%
Luca F., Pomerance C.
Colloquium Mathematicum
|
2012
|
tom 126
|
nr 1
2
Content available remote

On the distribution of some integers related to perfect and amicable numbers

100%
Pollack P., Pomerance C.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 2
169-182
EN
Let $s'(n) = ∑_{d|n, 1First, we show that the count of quasiperfect n ≤ x is at most $x^{1/4+o(1)}$ as x → ∞. In fact, we show that for each fixed a, there are at most $x^{1/4+o(1)}$ natural numbers n ≤ x with σ(n) ≡ a (mod n) and σ(n) odd. (Quasiperfect n satisfy these conditions with a = 1.) For fixed δ ≠ 0, define the arithmetic function $s_{δ}(n) := σ(n) - n - δ$. Thus, s₁ = s'. Our second theorem says that the number of n ≤ x which are amicable with respect to $s_{δ}$ is at most $x/(log x)^{1/2+o(1)}$.
3
Content available remote

The range of the sum-of-proper-divisors function

100%
Luca F., Pomerance C.
Acta Arithmetica
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2015
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tom 168
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nr 2
187-199
EN
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.
4
Content available remote

The iterated Carmichael λ-function and the number of cycles of the power generator

100%
Martin G., Pomerance C.
Acta Arithmetica
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2005
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tom 118
|
nr 4
305-335
5
Content available remote

On the periods of the linear congruential and power generators

100%
Kurlberg P., Pomerance C.
Acta Arithmetica
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2005
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tom 119
|
nr 2
149-169
6
Content available remote

On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ

100%
Luca F., Pomerance C.
Colloquium Mathematicum
|
2002
|
tom 92
|
nr 1
111-130
EN
Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n) on a set of asymptotic density 1.
7
Content available remote

On the range of Carmichael's universal-exponent function

100%
Luca F., Pomerance C.
Acta Arithmetica
|
2014
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tom 162
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nr 3
289-308
EN
Let λ denote Carmichael's function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler's φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x/(log x)^{.36}$ for all large x, while for φ it is equal to $x/(log x)^{1+o(1)}$, an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.
8
Content available remote

On the smallest pseudopower

81%
Bourgain J., Konyagin S., Pomerance C., Shparlinski I.
Acta Arithmetica
|
2009
|
tom 140
|
nr 1
43-55
9
Content available remote

Product-free sets with high density

81%
Kurlberg P., Lagarias J., Pomerance C.
Acta Arithmetica
|
2012
|
tom 155
|
nr 2
163-173
10
Content available remote

Odd perfect numbers are divisible by at least seven distinct primes

79%
Pomerance C.
Acta Arithmetica
|
1973-1974
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tom 25
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nr 3
265-300
11
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On the composition of the arithmetic functions σ and φ

69%
Pomerance C.
Colloquium Mathematicum
|
1989-1990
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tom 58
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nr 1
11-15
12
Content available remote

Carmichael's lambda function

60%
Erdős P., Pomerance C., Schmutz E.
Acta Arithmetica
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1991
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tom 58
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nr 4
363-385
13
Content available remote

On the number of distinct values of Euler's φ-function

60%
Maier H., Pomerance C.
Acta Arithmetica
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1987-1988
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tom 49
|
nr 3
263-275
14
Content available remote

On the congruences $σ(n) ≡ a (mod n)$ and $n ≡ a (mod φ(n))$

60%
Pomerance C.
Acta Arithmetica
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1974-1975
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tom 26
|
nr 3
265-272
15
Content available remote

On prime divisors of Mersenne numbers

50%
Erdős P., Kiss P., Pomerance C.
Acta Arithmetica
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1991
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tom 57
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nr 3
267-281
16
Content available remote

On the distribution of the values of Euler's function

50%
Pomerance C.
Acta Arithmetica
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1986
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tom 47
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nr 1
63-70
17
Content available remote

On primitive divisors of Mersenne numbers

40%
Pomerance C.
Acta Arithmetica
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1985-1986
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tom 46
|
nr 4
355-367
18
Content available remote

On composite n for which φ(n) | n -1

40%
Pomerance C.
Acta Arithmetica
|
1975-1976
|
tom 28
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nr 4
387-389
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