cp's OEIS Frontend

A247077 Composite numbers for which the harmonic mean of proper divisors is an integer.

%I A247077 #52 Feb 06 2021 10:22:39
%S A247077 1645,88473,63626653506
%N A247077 Composite numbers for which the harmonic mean of proper divisors is an integer.
%C A247077 Of course, for all prime numbers the harmonic mean of proper divisors is an integer.
%C A247077 a(4) >= 2*10^11. - _Hiroaki Yamanouchi_, Nov 20 2014
%C A247077 Conjecture: all terms are of the form m*(sigma(m)-1) where sigma(m)-1 is prime. - _Chai Wah Wu_, Dec 15 2020
%C A247077 a(4) <= 22351741783447265625. - _Daniel Suteu_, Dec 16 2020
%C A247077 From _Chai Wah Wu_, Feb 04 2021: (Start)
%C A247077 Other terms of the sequence of the form m*(sigma(m)-1) correspond to the following values of m:
%C A247077     3 *   5^143
%C A247077     3 *   5^623
%C A247077     3 *   5^1423
%C A247077     5 *   7^127
%C A247077     5 *   7^6595
%C A247077   101 * 103^25
%C A247077 (End)
%C A247077 Equivalently, composite numbers k such that sigma(k)-1 divides k*(tau(k)-1), where sigma = A000203 and tau = A000005. -  _Daniel Suteu_, Feb 05 2021
%e A247077 The proper divisors of 1645 are [1,5,7,35,47,235,329] and their harmonic mean is 7/(1/1 + 1/5 + 1/7 + 1/35 + 1/47 + 1/235 + 1/329) = 5.
%t A247077 Select[Range[2,100000],(IntegerQ[HarmonicMean[Most[Divisors[#]]]] && Not[PrimeQ[#]])&] (* _Daniel Lignon_, Nov 17 2014 *)
%o A247077 (PARI) lista(nn) = forcomposite (n=2, nn, my(d=divisors(n)); if (denominator((#d-1)/sum(i=1, #d-1, 1/d[i])) == 1, print1(n, ", "))); \\ _Michel Marcus_, Nov 17 2014
%o A247077 (PARI) isok(n) = n > 1 && !isprime(n) && (n*(numdiv(n)-1)) % (sigma(n)-1) == 0; \\ _Daniel Suteu_, Feb 05 2021
%Y A247077 Cf. A001599 for harmonic mean of all divisors and A247078 for harmonic mean of nontrivial divisors.
%K A247077 nonn,more,bref
%O A247077 1,1
%A A247077 _Daniel Lignon_, Nov 17 2014
%E A247077 a(3) from _Hiroaki Yamanouchi_, Nov 20 2014