A247077 -id:A247077 - cp's OEIS frontend

A247078 Numbers for which the harmonic mean of nontrivial divisors is an integer.

4, 9, 25, 49, 121, 169, 289, 345, 361, 529, 841, 961, 1050, 1369, 1645, 1681, 1849, 2209, 2809, 3481, 3721, 4386, 4489, 5041, 5329, 6241, 6489, 6889, 7921, 8041, 9409, 10201, 10609, 11449, 11881, 12769, 13026, 16129, 17161, 18769, 19321, 22201, 22801

Offset: 1

Daniel Lignon, Nov 17 2014

nonn

All the squares of prime numbers (A001248) have this property but there are other numbers (A247079): 345, 1050, 1645, 4386, 6489, 8041, ...

			The divisors of 25 are [1,5,25] and the nontrivial divisors are [5]. The harmonic mean is 1/(1/5)=5. That's the same for all squares of prime numbers.
The nontrivial divisors of 345 are [3,5,15,23,69,115] and their harmonic mean is 6/(1/3+1/5+1/15+1/23+1/69+1/115) = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1269 from Daniel Lignon)

Cf. similar sequences: A001599 (with all divisors), A247077 (with proper divisors).

hm:= S -> nops(S)/convert(map(t->1/t,S),`+`):
filter:= n -> not isprime(n) and type(hm(numtheory:-divisors(n) minus {1,n}),integer):
select(filter, [$2..10^5]); # Robert Israel, Nov 17 2014

Select[Range[2,100000],(Not[PrimeQ[#]] && IntegerQ[HarmonicMean[Rest[Most[Divisors[#]]]]])&]

isok(n) = my(d=divisors(n)); (#d >2) && (denominator((#d-2)/sum(i=2, #d-1, 1/d[i])) == 1);

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

Original entry on oeis.org

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

Amiram Eldar, May 29 2020

nonn

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.

			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.

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

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.

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 *)

A335270 Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.

Original entry on oeis.org

228, 1645, 7725, 88473, 20295895122, 22550994580

Offset: 1

Amiram Eldar, May 29 2020

Since 1 is the only proper unitary divisor of powers of prime (A000961), they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 4, 5, 5, 9, 18, 20.
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-1) | m*(2^omega(m)-1), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) - 1 = A034444(m) - 1 = A309307(m) is the number of the proper unitary divisors of m.
The squarefree terms of A247077 are also terms of this sequence.
a(7) > 10^12, if it exists. - Giovanni Resta, May 30 2020
Conjecture: all terms are of the form n*(usigma(n)-1) where usigma(n)-1 is prime. - Chai Wah Wu, Dec 17 2020

			228 is a term since the harmonic mean of its proper unitary divisors, {1, 3, 4, 12, 19, 57, 76} is 4 which is an integer.

The unitary version of A247077.

Cf. A006086, A000961, A024619, A034444, A034448, A077610, A309307, A335268, A335269.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^5], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - 1] &]

a(5)-a(6) from Giovanni Resta, May 30 2020

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A247078 Numbers for which the harmonic mean of nontrivial divisors is an integer.

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

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A335270 Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.

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