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 A247077 Composite numbers for which the harmonic mean of proper divisors is an integer. 3
 1645, 88473, 63626653506 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Of course, for all prime numbers the harmonic mean of proper divisors is an integer. a(4) >= 2*10^11. - Hiroaki Yamanouchi, Nov 20 2014 Conjecture: all terms are of the form m*(sigma(m)-1) where sigma(m)-1 is prime. - Chai Wah Wu, Dec 15 2020 a(4) <= 22351741783447265625. - Daniel Suteu, Dec 16 2020 From Chai Wah Wu, Feb 04 2021: (Start) Other terms of the sequence of the form m*(sigma(m)-1) correspond to the following values of m:     3 *   5^143     3 *   5^623     3 *   5^1423     5 *   7^127     5 *   7^6595   101 * 103^25 (End) 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 LINKS EXAMPLE 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. MATHEMATICA Select[Range[2, 100000], (IntegerQ[HarmonicMean[Most[Divisors[#]]]] && Not[PrimeQ[#]])&] (* Daniel Lignon, Nov 17 2014 *) PROG (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 (PARI) isok(n) = n > 1 && !isprime(n) && (n*(numdiv(n)-1)) % (sigma(n)-1) == 0; \\ Daniel Suteu, Feb 05 2021 CROSSREFS Cf. A001599 for harmonic mean of all divisors and A247078 for harmonic mean of nontrivial divisors. Sequence in context: A135016 A238066 A349178 * A093059 A255620 A253030 Adjacent sequences:  A247074 A247075 A247076 * A247078 A247079 A247080 KEYWORD nonn,more,bref AUTHOR Daniel Lignon, Nov 17 2014 EXTENSIONS a(3) from Hiroaki Yamanouchi, Nov 20 2014 STATUS approved

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Last modified November 29 18:41 EST 2021. Contains 349416 sequences. (Running on oeis4.)