A lower bound for odd triperfects

The energy of unitary cayley graphs

For a natural number \(n\), let \(A(n)\) denote the smallest natural number that has exactly \(n\) divisors. Let \(p_{1}p_{2}\ldots p_{k}\) be the prime decomposition of \(n\) where the primes are given in decreasing order and the factors are not necessarily distinct. If \(q_{1},\ldots ,q_{k}\) denote the first \(k\) primes and \(A(n)=q_{1}^{p_{1}-1}\ldots q_{k}^{p_{k}-1}\), we say that \(n\) is ordinary. If \(n\) is not ordinary, we say it is extraordinary. In Brown (2006), it was shown that almost all numbers are ordinary and if \(E_{x}\) denotes the set of extraordinary numbers less than or equal to a positive real number \(x\), \(|E_{x}|=o\left( \frac{x}{2^{(\log (\log x))^{\delta }}}\right) \) for any \(\displaystyle 0<\delta <1/2\). In what follows, we will prove that \(\displaystyle |E_x|=\frac{x}{\log x}e^{\Psi (x)}\) where \(\displaystyle \Psi (x)\sim \frac{1}{2\log 2}\frac{\log _{(3)}^2x}{\log _{(4)}x}\) and \(\displaystyle \log _{(r)}x\) denotes the \(r\)-times iterated natural logarithm. We make use of the Prime Number Theorem as well as estimates of the size of sets where the number of total prime factors or distinct prime factors is fixed.

For a positive integer [Formula: see text], if [Formula: see text] denotes the sum of the positive divisors of [Formula: see text], then [Formula: see text] is called a deficient perfect number if [Formula: see text] for some positive divisor [Formula: see text] of [Formula: see text]. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

An odd perfect number, N N , is shown to have at least nine distinct prime factors. If 3 ∤ N 3\nmid N then N N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.

Let R m be endowed with the Euclidean metric. The covering radius of a lattice Λ ⊂ R m is the least distance r such that, given any point of R m , the distance from that point to Λ is not more than r . Lattices can occur via the unit group of the ring of integers in an algebraic number field K , by applying a logarithmic embedding K ⁎ → R m . In this paper, we examine those lattices which arise from the cyclotomic number field Q ( ζ n ) , for a given positive integer n ≥ 5 such that n ≢ 2 ( mod 4 ) . We then provide improvements to a result of de Araujo in [3] , and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2] , and show that, asymptotically, it can be improved no further.

Let be a positive integer and be the sum of the digits in basis q of the positive integer n. We prove that the quotient has a normal order one, where and are respectively, the number of distinct prime factors and the number of prime factors p of a positive integer n counted with multiplicity such that mod Moreover, we discuss sums of the form where f is a multiplicative function.

We make a table of odd integers N with five distinct prime factors for which 2 − 10 − 12 &gt; σ ( N ) / N &gt; 2 + 10 − 12 2 - {10^{ - 12}} &gt; \sigma (N)/N &gt; 2 + {10^{ - 12}} , and show that for such N | σ ( N ) / N − 2 | &gt; 10 − 14 N\;|\sigma (N)/N - 2| &gt; {10^{ - 14}} . Using this inequality, we prove that there are no odd perfect numbers, no quasiperfect numbers and no odd almost perfect numbers with five distinct prime factors. We also make a table of odd primitive abundant numbers N with five distinct prime factors for which 2 &gt; σ ( N ) / N &gt; 2 + 2 / 10 10 2 &gt; \sigma (N)/N &gt; 2 + 2/{10^{10}} .

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

A fundamental result, due to Sabidussi and Vizing, states that every connected graph has a unique prime factorization rel- ative to the Cartesian product; but disconnected graphs are not uniquely prime factorable. This paper describes a system of modular arithmetic on graphs under which both connected and disconnected graphs have unique prime Cartesian factorizations. The Cartesian product of two simple graphs G = (V (G); E(G)) and H = (V (H); E(H)) is the graph G H with V (G H) = V (G) V (H), and (u; x)(v; y) 2 E(G H) if either u = v and xy 2 E(H), or uv 2 E(G) and x = y. This product is commutative and associative: G H = H G and G (H K) = (G H) K (up to isomorphism) for all graphs G, H and K. Also G H is connected if and only if both G and H are connected. For a full treatment of this product, see Chapter 4 of Imrich and Kla zar (2). We denote the empty graph (i.e. the graph with no vertices) as O, and the complete graph on n vertices as Kn. Notice that G O = O and G K1 = G for all graphs G. If n 2 N, then nG denotes the graph that is the disjoint union of n copies of G (or O if n = 0). Note n(G H) = nG H = G nH. For a positive integer n, we dene G n = G G G (n factors) and we adopt the convention G 0 = K1. A graph G is prime if it is nontrivial and G = G1 G2 implies G1 = K1 or G2 = K1. Every graph G has a prime factorization G = G1 G2 Gp, where each factor Gi is prime. A fundamental theorem, proved indepen- dently by Sabidussi (3) and Vizing (4) states that the prime factorization of a connected graph is unique, that is if a connected graph G has prime factor- izations G1 G2 Gp and H1 H2 Hq, then p = q and Gi = Hi for 1 i p (after reindexing, if necessary). But disconnected graphs are not uniquely prime factorable, in general. One standard example is the graph G = K1 + K2 + K 2

A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4k2.

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three “sporadic” groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.

In this paper we prove that odd integers n with (n) = 2(n + 1) have at least six distinct prime factors.

On the modulo degree complexity of Boolean functions

A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.

An almost-prime sieve