cp's OEIS Frontend

A335267 Composite numbers whose harmonic mean of their divisors that are larger than 1 is an integer.

%I A335267 #16 Jan 22 2021 06:03:02
%S A335267 6,15,28,30,91,117,135,252,270,496,703,864,936,1891,1989,2295,2701,
%T A335267 4284,4590,5733,8128,8432,12403,18721,19872,21528,38503,41580,49141,
%U A335267 51319,56896,79003,88831,104653,121920,146611,188191,218791,226801,235053,269011,286903
%N A335267 Composite numbers whose harmonic mean of their divisors that are larger than 1 is an integer.
%C A335267 The primes are excluded from this sequence since they are trivial terms.
%C A335267 The corresponding harmonic means are 3, 5, 5, 5, 13, 9, 9, 9, 9, 9, 37, ...
%C A335267 Equivalently, composite numbers m such that (sigma(m)-m) | m*(tau(m)-1), or A001065(m) | A168014(m).
%C A335267 The semiprimes terms of this sequence are of the form p*q where p and q = 2*p - 1 are primes (A129521).
%C A335267 If m is a k-perfect numbers, k = 2, 3, ... (i.e., sigma(m) = k*m), then sigma(m)-m = (k-1)*m. If (k-1)*m | m*(tau(m)-1) then (k-1) | (tau(m)-1). If k is odd then tau(m) is also odd, so m is a square, and sigma(m) is odd. Since m | sigma(m) this means that m is also odd. Since there is no known odd multiply-perfect number except for 1 (A007691), there are no known k-perfect numbers with odd k in this sequence.
%C A335267 The perfect numbers (k=2, A000396) are terms: if m is a perfect number then sigma(m)-m = m.
%C A335267 The 4-perfect number (k=4, A027687) m are terms if 3 | (tau(m)-1). Of the first 36 terms of A027687 there are 8 such terms, the first is A027687(26).
%C A335267 The 6-perfect number (k=6, A046061) m are terms if 5 | (tau(m)-1). Of the first 245 terms of A046061 there are 20 such terms, the first is A046061(19).
%C A335267 Hemiperfect numbers that are terms of this sequence include A055153(i) for i = 10, 18 and 20, A141645(21), and A159271(i) for i = 97 and 103.
%H A335267 Amiram Eldar, <a href="/A335267/b335267.txt">Table of n, a(n) for n = 1..1000</a>
%e A335267 6 is a term since its divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.
%t A335267 Select[Range[10^6], CompositeQ[#] && Divisible[# * (DivisorSigma[0, #] - 1), DivisorSigma[1, #] - #] &]
%t A335267 Select[Range[287000],CompositeQ[#]&&IntegerQ[HarmonicMean[ Rest[ Divisors[ #]]]]&] (* _Harvey P. Dale_, Jan 21 2021 *)
%Y A335267 A000396 and A129521 are subsequences.
%Y A335267 Similar sequences: A001599, A247077, A247078.
%Y A335267 Cf. A000005 (tau), A000203 (sigma).
%Y A335267 Cf. A001065, A032741, A168014,
%Y A335267 Cf. A005382, A005383.
%Y A335267 Cf. A027687, A046061, A055153, A141645, A159271.
%K A335267 nonn
%O A335267 1,1
%A A335267 _Amiram Eldar_, May 29 2020