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Integers m such that A240471(m) > A115588(m).
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%I #67 Dec 26 2020 15:58:37

%S 16,24,27,28,32,36,44,48,50,52,54,55,60,64,65,68,72,76,77,80,81,84,85,

%T 90,91,92,95,96,98,100,105,108,110,112,115,116,119,120,124,125,126,

%U 128,130,132,133,135,136,140,143,144,145,148,150,152,154,155,156,160

%N Integers m such that A240471(m) > A115588(m).

%C 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.

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

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

%C 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.

%C 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.

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

%C 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.

%C 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.

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

%o (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;}

%o a8(n) = {my(f=factor(n), list=List()); #select(isprime, Set(Vec(listf(f, list))));}

%o a1(n) = n*numdiv(n)\sigma(n);

%o isok(m) = a1(m) > a8(m); \\ _Michel Marcus_, Dec 02 2020

%Y Cf. A240471, A115588, A322448.

%K nonn

%O 1,1

%A _Thomas Scheuerle_, Nov 30 2020