In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the ...network increases, the network will be connected whenever the largest possible node index n≥49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6/π2, whereas the average degree increases linearly with slope 6/π2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49≤n≤288 and it is at most 2 for n≥289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r-length labeled cycles, which indicates the existence of a cycle of length at most O(logn). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási–Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator–prey networks which can easily avoid the global synchronization.
In this paper, we study the asymptotic behavior of the counting function of composite numbers with three constrained prime factors. Our constraint is that the largest prime factor is bounded above by ...a general increasing function with respect to either the smallest or the second smallest prime factor.
We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: κx(n)=nx+∑d⌊nκx(d), where the sum is over the proper divisors of n. We give a geometrical ...interpretation of κx(n), which we use to derive a relation between κx(n) and κ0(n). For x≥2, we observe that κx(n)/nx<1/(2−ζ(x)). We show that, for n≥2, κ0(n) is twice the number of ordered factorizations, a problem much studied in its own right. By computing those numbers that are more recursively divisible than all of their predecessors, we recover many of the numbers prevalent in design and technology, and suggest new ones which have yet to be adopted.
Based on an arithmetical and autocatalytic approach, the authors propose a solution for the occurrence of prime numbers. Exact arithmetical calculations are provided for: the closest prime to any ...given positive integer (or any number of bigger or smaller primes from that integer); the quantity of prime (and composite) numbers between 1 and any positive integer; the quantity of prime (and composite) numbers between any two positive integers.
Based on an arithmetical and autocatalytic approach, the authors propose a solution for the occurrence of prime numbers. Exact arithmetical calculations are provided for: the closest prime to any ...given positive integer (or any number of bigger or smaller primes from that integer); the quantity of prime (and composite) numbers between 1 and any positive integer; the quantity of prime (and composite) numbers between any two positive integers.
Dans cette note, nous prouvons que pour tout entier fixé K ≥ 2, pour tout ϵ > 0 et pour tout x suffisamment grand, il existe au moins x
1 − ϵ
nombres premiers x < p ≤ (1 + K
−1)x tels que tous les ...nombres entiers de la forme pj ± ahk avec 2 ≤ a ≤ K < |K| ≤ K, 1 ≤ j ≤ K, 0 ≤ h ≤ K ≤ log x sont des nombres composés ayant au moins (log log x)1 − ϵ
facteurs premiers distincts.
In this note, we prove that for any fixed integer K ≥ 2, for all ϵ > 0 and for all sufficiently large x, there exist at least x
1 − ϵ
primes x < p ≤ (1 + K
−1)x, such that all of the integers pj ± ahk avec 2 ≤ a ≤ K < |K| ≤ K, 1 ≤ j ≤ K, 0 ≤ h ≤ K ≤ log x are composite having at least (log log x)1 − ϵ
distinct prime factors.
Let us denote by
d
(
n
) the number of divisors of
n
, by
li
(
t
)
the logarithmic integral of
t
, by
β
2
the number
log
3
/
2
log
2
=
0.584
…
and by
R
(
t
) the function
t
↦
2
t
+
∑
ρ
t
ρ
/
ρ
2
log
...2
t
, where
ρ
runs over the non-trivial zeros of the Riemann
ζ
function. In his PHD thesis about highly composite numbers, Ramanujan proved, under the Riemann hypothesis, that
log
d
(
n
)
log
2
⩽
li
(
log
n
)
+
β
2
li
(
log
β
2
n
)
-
log
β
2
n
log
log
n
-
R
(
log
n
)
+
O
log
n
(
log
log
n
)
3
holds when
n
tends to infinity. The aim of this paper is to give an effective form to the above asymptotic result of Ramanujan.
In this note, we show that, for any real number τ∈12,1), any finite set of positive integers K and any integer s1≥2, the sequence of integers s1,s2,s3,… satisfying si+1−si∈K if si is a prime number, ...and 2≤si+1≤τsi if si is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on τ,s1 and the maximal element of K only. In particular, if K is a singleton set and for each composite si the integer si+1 in the interval 2,τsi is chosen by some prescribed rule, e.g., si+1 is the largest prime divisor of si, then the sequence s1,s2,s3,… is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either K={1} and τ∈12,34) or K={2} and τ∈12,59).
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