A006086 -id:A006086 - cp's OEIS frontend

A335269 Numbers for which the harmonic mean of the nontrivial unitary divisors is an integer.

228, 345, 1645, 2120, 4025, 4386, 4977, 7725, 8041, 13026, 23881, 24157, 24336, 51925, 88473, 115957, 150161, 169893, 229177, 255041, 278721, 322592, 342637, 377201, 490725, 538625, 656937, 1497517, 1566981, 2132021, 3256261, 3847001, 4646101, 5054221, 5524897

Offset: 1

Amiram Eldar, May 29 2020

nonn

A number m is a term if the set {d|m ; d > 1, d < m, gcd(d, m/d) = 1} is nonempty and the harmonic mean its members is an integer.
The corresponding harmonic means are 8, 9, 15, 16, 25, 12, 21, 15, 33, 12, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m-1) | m*(2^omega(m)-2), where usigma is the sum of unitary divisors (A034448), and 2^omega(m)-2 = A034444(m)-2 = A087893 (m) is the number of the nontrivial unitary divisors of m.
The squarefree terms of A247078 are also terms of this sequence.

			228 is a term since the harmonic mean of its nontrivial unitary divisors, {3, 4, 12, 19, 57, 76} is 8 which is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..150

The unitary version of A247078.

Cf. A006086, A034444, A034448, A077610, A087893, A335268, A335270.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^6], (omega = PrimeNu[#]) > 1 && Divisible[#*(2^omega - 2), usigma[#] - # - 1] &]

A353039 Unitary arithmetic numbers k whose mean unitary divisor is a unitary divisor of k.

Original entry on oeis.org

1, 6, 60, 420, 630, 5460, 8190, 16632, 64260, 143640, 172900, 598500, 716625, 790398, 791700, 1182384, 1187550, 1530144, 2708160, 4277448, 5314680, 6284250, 6397300, 6741630, 14619150, 15214500, 22144500, 24315984, 87966648, 93284100, 161670600, 165197760, 232517250

Offset: 1

Amiram Eldar, Apr 19 2022

nonn

Also, unitary harmonic numbers k whose harmonic mean of the unitary divisors of k is a unitary divisor of k.

			6 is a term since the arithmetic mean of its unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a unitary divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..84

Subsequence of A006086 and A103826.

Cf. A007340.

q[n_] := Module[{f = FactorInteger[n], d, s, m}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); m = s/d; IntegerQ[m] && Divisible[n, m] && CoprimeQ[m, n/m]]; Select[Range[10^6], q]

A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 45, 48, 56, 60, 64, 90, 96, 112, 120, 128, 180, 192, 224, 240, 256, 360, 384, 448, 480, 496, 512, 720, 768, 896, 960, 992, 1024, 1440, 1536, 1792, 1920, 1984, 2048, 2880, 3072, 3584, 3840, 3968, 4096, 5760, 6144, 7168, 7680

Offset: 1

Amiram Eldar, May 04 2023

Equivalently, the set of divisors can be defined by {d | k, BitAnd(k, d) = d}.
Analogous to harmonic (or Ore) numbers (A001599) where the divisors d of k are restricted by BitOr(k, d) = k or BitAnd(k, d) = d.
If k is a term then so is 2*k. The primitive terms are in A362805. Thus, this sequence includes all the powers of 2 (A000079), all the numbers of the form 3*2^m and 15*2^m for m >= 1, and all the numbers of the form 7*2^m for m >= 2.
All the even perfect numbers (A000396) are terms: if k = 2^(p-1)*(2^p-1) is a perfect number (where p is a Mersenne exponent, A000043), then the only divisors of k such that BitOr(k, d) = k are 2^(p-1) and k itself, and the harmonic mean of 2^(p-1) and 2^(p-1)*(2^p-1) is 2^p - 1.
Are 1 and 45 the only odd terms in this sequence?

Amiram Eldar, Table of n, a(n) for n = 1..406

Cf. A000043, A000396, A246600, A246601.
Subsequences: A000079, A007283 \ {3}, A005009 \ {7, 14}, A110286 \ {15}, A362805.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]

div(n) = select(x->(bitor(x, n) == n), divisors(n));
is(n) = {my(d = div(n)); denominator(#d/sum(i = 1, #d, 1/d[i])) == 1;}

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

A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 2970, 5460, 8190, 9100, 15925, 27300, 36720, 40950, 46494, 47520, 54600, 81900, 95550, 136500, 163800, 172900, 204750, 232470, 245700, 257040, 332640, 409500, 464940, 491400, 646425, 716625, 790398, 791700, 819000, 900900, 929880

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Equivalently, numbers k such that A324706(k) | (k * A335385(k)).

Differs from A063947 from n >= 18.

			45 is a term since its tri-unitary divisors are {1, 5, 9, 45} and their harmonic mean, 3, in an integer.

Amiram Eldar, Table of n, a(n) for n = 1..200

A324707 is a subsequence.
Analogous sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary).
Cf. A324706, A335385.

f1[p_, e_] := If[e == 3 || e == 6, 4, 2]; f2[p_, e_] := If[e == 3, (p^4 - 1)/(p - 1), If[e == 6, (p^8 - 1)/(p^2 - 1), p^e + 1]]; f[p_, e_] := p^e * f1[p, e]/f2[p, e]; tuhQ[1] = True; tuhQ[n_] := IntegerQ[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^4], tuhQ]

A348918 Noninfinitary harmonic numbers: numbers such that the harmonic mean of their noninfinitary divisors is an integer.

Original entry on oeis.org

4, 9, 12, 18, 25, 45, 49, 60, 96, 112, 121, 126, 150, 169, 289, 294, 336, 361, 448, 486, 529, 540, 560, 600, 637, 672, 726, 841, 961, 1014, 1232, 1344, 1350, 1369, 1638, 1680, 1681, 1734, 1849, 2166, 2209, 2430, 2809, 2850, 3174, 3481, 3721, 3822, 4200, 4320, 4489

Offset: 1

Amiram Eldar, Nov 04 2021

nonn

Includes all the squares of primes (A001248), since they are the numbers with a single noninfinitary divisor.

			12 is a term since its noninfinitary divisors are {2, 6}, and their harmonic mean, 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A001248, A001599, A006086, A063947, A286325, A319745.

nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; Select[Range[5000], (d = nidiv[#]) != {} && IntegerQ@ HarmonicMean[d] &]

A361384 a(n) is the number of distinct prime factors of the n-th unitary harmonic number.

Original entry on oeis.org

0, 2, 2, 3, 3, 4, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 4, 3, 5, 4, 5, 5, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 4, 4, 4, 5, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 6, 5, 5, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 5, 6, 5, 6, 5, 6, 5, 6, 4, 5, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 6, 5, 6

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Each term appears a finite number of times in the sequence (Hagis and Lord, 1975).

Amiram Eldar, Table of n, a(n) for n = 1..290
Peter Hagis, Jr. and Graham Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., Vol. 51, No. 1 (1975), pp. 1-7.

Cf. A001221, A006086, A006087.

uh[n_] := n * Times @@ (2/(1 + Power @@@ FactorInteger[n])); uh[1] = 1; PrimeNu[Select[Range[10^6], IntegerQ[uh[#]] &]]

uhmean(n) = {my(f = factor(n)); n*prod(i=1, #f~, 2/(1+f[i, 1]^f[i, 2])); };
lista(kmax) = {my(uh); for(k = 1, kmax, uh = uhmean(k); if(denominator(uh) == 1, print1(omega(k), ", ")));}

a(n) = A001221(A006086(n)).

A317477 Unitary harmonic numbers m such that the harmonic mean of the unitary divisors of m is also a unitary harmonic number.

Original entry on oeis.org

1, 1512, 624937680, 808742880, 87348127500, 139103775360, 150780265560, 261880857000

Offset: 1

Amiram Eldar, Jul 29 2018

The harmonic means of the unitary divisors of the terms are 1, 6, 45, 45, 45, 90, 45, 90.

			1512 is in the sequence since the harmonic mean of its divisors is 6 and 6 is also a unitary harmonic number.

Cf. A006086, A317476.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; ud[n_] := 2^PrimeNu[n]; uhQ[n_] := IntegerQ[n*ud[n]/usigma[n]]; uhuhQ[n_] :=  Module[{m = n*ud[n]/usigma[n]}, IntegerQ[m] && uhQ[m]]; Do[If[uhuhQ[n], Print[n]], {n, 1, 10^11}]

A332477 Numbers k that are unitary harmonic in Gaussian integers: k * A332476(k) is divisible by A332472(k) + i*A332473(k) (where i is the imaginary unit).

Original entry on oeis.org

1, 5, 12, 50, 60, 84, 300, 420, 450, 756, 900, 1950, 3780, 7800, 9900, 33150, 49140, 54600, 100800, 132600, 265200, 491400, 928200, 1856400, 8353800, 8884200, 16707600, 52211250, 65995776, 78566400, 182739375, 183783600, 208845000, 280348992, 293046000, 329978880

Offset: 1

Amiram Eldar, Feb 13 2020

nonn

Analogous to unitary harmonic numbers (A006086), with the number and sum of unitary divisors functions generalized for Gaussian integers (A332476, A332472 + i * A332473) instead of the number and sum of unitary divisors functions (A034444, A034448).

			5 is a term since 5 * A332476(5)/(A332472(5) + i*A332473(5)) = 5 * 4/(4 + 8*i) = 1 - 2*i is a Gaussian integer.

Amiram Eldar, Table of n, a(n) for n = 1..75

Cf. A034444, A034448, A006086, A332317, A332472, A332473, A332476.

sigma[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; tau[p_, e_] := If[Abs[p] == 1, 1, 2]; unitaryHarmonicQ[n_] := Divisible[n * Times @@ tau @@@ (f = FactorInteger[n, GaussianIntegers -> True]), Times @@ sigma @@@ f]; Select[Range[10^5], unitaryHarmonicQ]

A348923 Numbers that are both unitary and nonunitary harmonic numbers.

Original entry on oeis.org

45, 60, 3780, 64260, 3112200, 6320160

Offset: 1

Amiram Eldar, Nov 04 2021

a(7) > 10^12, if it exists.
For each term the two sets of unitary and nonunitary divisors both contain more than one element. The only number with a single unitary divisor is 1 which does not have nonunitary divisors. Numbers with a single nonunitary divisor are the squares of primes which are not unitary harmonic numbers. Therefore, this sequence is a subsequence of A348715.
Nonsquarefree numbers k such that A034448(k) divides k*A034444(k) and A048146(k) divides k*A048105(k). - Daniel Suteu, Nov 05 2021

			45 is a term since the unitary divisors of 45 are 1, 5, 9 and 45, and their harmonic mean is 3, and the nonunitary divisors of 45 are 3 and 15, and their harmonic mean is 5.

Intersection of A006086 and A319745.
Subsequence of A348715.
Cf. A348922.
Cf. A034444, A034448, A048105, A048146.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[65000], !SquareFreeQ[#] && IntegerQ[# * (d = 2^PrimeNu[#])/ (s = usigma[#])] && IntegerQ[# * (DivisorSigma[0, #] - d)/(DivisorSigma[1, #] - s)] &]

cp's OEIS Frontend

A335269 Numbers for which the harmonic mean of the nontrivial unitary divisors is an integer.

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A353039 Unitary arithmetic numbers k whose mean unitary divisor is a unitary divisor of k.

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A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

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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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A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

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A348918 Noninfinitary harmonic numbers: numbers such that the harmonic mean of their noninfinitary divisors is an integer.

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A361384 a(n) is the number of distinct prime factors of the n-th unitary harmonic number.

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A317477 Unitary harmonic numbers m such that the harmonic mean of the unitary divisors of m is also a unitary harmonic number.

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