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

6, 15, 28, 30, 91, 117, 135, 252, 270, 496, 703, 864, 936, 1891, 1989, 2295, 2701, 4284, 4590, 5733, 8128, 8432, 12403, 18721, 19872, 21528, 38503, 41580, 49141, 51319, 56896, 79003, 88831, 104653, 121920, 146611, 188191, 218791, 226801, 235053, 269011, 286903

Offset

1,1

Comments

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

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

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

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

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.

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

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).

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).

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Example

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.

Mathematica

Select[Range[10^6], CompositeQ[#] && Divisible[# * (DivisorSigma[0, #] - 1), DivisorSigma[1, #] - #] &]
Select[Range[287000], CompositeQ[#]&&IntegerQ[HarmonicMean[ Rest[ Divisors[ #]]]]&] (* Harvey P. Dale, Jan 21 2021 *)

Crossrefs

A000396 and A129521 are subsequences.

Similar sequences: A001599, A247077, A247078.

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

Cf. A001065, A032741, A168014,

Cf. A005382, A005383.

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

Sequence in context: A165454 A333646 A063525 * A349476 A242454 A161777

Adjacent sequences: A335264 A335265 A335266 * A335268 A335269 A335270

Keyword

nonn

Author

Amiram Eldar, May 29 2020

Status

approved