A335267 - OEIS

%I #16 Jan 22 2021 06:03:02

%S 6,15,28,30,91,117,135,252,270,496,703,864,936,1891,1989,2295,2701,

%T 4284,4590,5733,8128,8432,12403,18721,19872,21528,38503,41580,49141,

%U 51319,56896,79003,88831,104653,121920,146611,188191,218791,226801,235053,269011,286903

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

%C The primes are excluded from this sequence since they are trivial terms.

%C The corresponding harmonic means are 3, 5, 5, 5, 13, 9, 9, 9, 9, 9, 37, ...

%C Equivalently, composite numbers m such that (sigma(m)-m) | m*(tau(m)-1), or A001065(m) | A168014(m).

%C The semiprimes terms of this sequence are of the form p*q where p and q = 2*p - 1 are primes (A129521).

%C 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 The perfect numbers (k=2, A000396) are terms: if m is a perfect number then sigma(m)-m = m.

%C 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 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 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 Amiram Eldar, <a href="/A335267/b335267.txt">Table of n, a(n) for n = 1..1000</a>

%e 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 Select[Range[10^6], CompositeQ[#] && Divisible[# * (DivisorSigma[0, #] - 1), DivisorSigma[1, #] - #] &]

%t Select[Range[287000],CompositeQ[#]&&IntegerQ[HarmonicMean[ Rest[ Divisors[ #]]]]&] (* _Harvey P. Dale_, Jan 21 2021 *)

%Y A000396 and A129521 are subsequences.

%Y Similar sequences: A001599, A247077, A247078.

%Y Cf. A000005 (tau), A000203 (sigma).

%Y Cf. A001065, A032741, A168014,

%Y Cf. A005382, A005383.

%Y Cf. A027687, A046061, A055153, A141645, A159271.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 29 2020