A339328 Integers m such that A240471(m) > A115588(m).

16, 24, 27, 28, 32, 36, 44, 48, 50, 52, 54, 55, 60, 64, 65, 68, 72, 76, 77, 80, 81, 84, 85, 90, 91, 92, 95, 96, 98, 100, 105, 108, 110, 112, 115, 116, 119, 120, 124, 125, 126, 128, 130, 132, 133, 135, 136, 140, 143, 144, 145, 148, 150, 152, 154, 155, 156, 160

Offset: 1

Author

Thomas Scheuerle, Nov 30 2020

Keywords

• nonn

Comments

Integers m such that integer part of the harmonic mean of divisors of m is greater than the number of distinct prime numbers necessary to represent m.

For all m not in this sequence this integer part is equal to the number of distinct prime numbers necessary to represent m.

This correlation between A240471 and A115588 contains some apparently random component.

If the integer part of the harmonic mean of divisors of m equals 1 we will find an 1 in A115588(m) too, for all m. If the integer part of the harmonic mean of divisors of m equals 2 we will find 2 in A115588(m) too, with probability of ~0.9877 for m in range 2-1000.

For m until 10000 the only exceptions are 16 and 27. If the integer part of the harmonic mean of divisors of m equals 3 we will find 3 in A115588(m) too, with probability of ~0.1983 for m in range 2-1000. For integer parts greater than 3 the probability gets fast smaller.

If m is a square of a prime it is not in this sequence.

Let m be a semiprime with two distinct prime factors p1 and p2. If m >= 3(1+p1+p2) then m is in this sequence. Example: 55 > 3(1+5+11). This can be generalized for k-almostprimes if all factors are distinct: If m(2^k) >= (1+k)sigma(m) then m is in this sequence. Example: 105*8 > 4*192.

Let p be a prime greater than 2. Let o be a natural number >0 without divisor p, then if m = o*p^p, m is in this sequence. This can be generalized for a set of distinct primes >2 {p_1,p_2,...,p_n} and any permutation of this set {p_a,p_b,...,p_z}, then if m = o*p_1^p_a*p_2^p_b*...*p_n^p_z, m is in this sequence. Example: 3960 = 55*2^3*3^2.

The sequence includes all numbers whose prime factorization contains at least one composite exponent (A322448).

Crossrefs

Cf. A240471, A115588, A322448.

Programs

PARI

listf(f, list) = {for (k=1, #f~, listput(list, f[k,1]); if (isprime(f[k,2]), listput(list, f[k,2]), if (f[k,2] > 1, my(vexp = Vec(listf(factor(f[k,2]), list))); for (i=1, #vexp, listput(list, vexp[i]););););); list;}
a8(n) = {my(f=factor(n), list=List()); #select(isprime, Set(Vec(listf(f, list))));}
a1(n) = n*numdiv(n)\sigma(n);
isok(m) = a1(m) > a8(m); \\ Michel Marcus, Dec 02 2020