A006087 -id:A006087 - cp's OEIS frontend

A006086 Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).

1, 6, 45, 60, 90, 420, 630, 1512, 3780, 5460, 7560, 8190, 9100, 15925, 16632, 27300, 31500, 40950, 46494, 51408, 55125, 64260, 66528, 81900, 87360, 95550, 143640, 163800, 172900, 185976, 232470, 257040, 330750, 332640, 464940, 565488, 598500, 646425, 661500

Offset: 1

N. J. A. Sloane

Let ud(n) and usigma(n) be number of and sum of unitary divisors of n; then the unitary harmonic mean of the unitary divisors is H(n) = n*ud(n)/usigma(n). - Emeric Deutsch, Dec 22 2004

A103340(a(n)) = 1; A103339(a(n)) = A006087(n). - Reinhard Zumkeller, Mar 17 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..290 (terms < 10^12)
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
P. Hagis, Jr. and G. Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7.
P. Hagis, Jr. and G. Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7. (Annotated scanned copy)
Charles R. Wall, Unitary harmonic numbers, Fibonacci Quarterly, Vol. 21, No. 1 (1983), pp. 18-25.

See A006087 for more info.

Cf. A103339, A103340.

a006086 n = a006086_list !! (n-1)
a006086_list = filter ((== 1) . a103340) [1..]
-- Reinhard Zumkeller, Mar 17 2012

ud[n_] := 2^PrimeNu[n]; usigma[n_] := Sum[ If[ GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; uhm[n_] := n*ud[n]/usigma[n]; Reap[ Do[ If[ IntegerQ[uhm[n]], Print[n]; Sow[n]], {n, 1, 10^6}]][[2, 1]] (* Jean-François Alcover, May 16 2013 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
isok(n) = my(v=udivs(n)); denominator(n*#v/vecsum(v))==1; \\ Michel Marcus, May 07 2017

is(n,f=factor(n))=(n<<(#f~))%sumdivmult([n,f], d, if(gcd(d, n/d)==1, d))==0 \\ Charles R Greathouse IV, Nov 05 2021

list(lim)=my(v=List()); forfactored(n=1,lim\1, if((n[1]<Charles R Greathouse IV, Nov 05 2021

If m is a term and omega(m) = A001221(m) = k, then m < 2^(k*2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

More terms from Emeric Deutsch, Dec 22 2004

A103339 Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 16, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 8, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 80, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 28, 57, 116, 59, 4, 61, 31, 63, 128, 65, 11, 67, 136, 23, 35, 71, 16, 73

Offset: 1

Emeric Deutsch, Jan 31 2005

			1, 4/3, 3/2, 8/5, 5/3, 2, ...
a(8) = 16 because the unitary divisors of 8 are {1,8} and 2/(1/1 + 1/8) = 16/9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A103340 (denominators), A099377, A099378.

Cf. A005117, A034444, A034448, A077610.

import Data.Ratio ((%), numerator)
a103339 = numerator . uhm where uhm n = (n * a034444 n) % (a034448 n)
-- Reinhard Zumkeller, Mar 17 2012

with(numtheory): udivisors:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k],n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end: utau:=n->nops(udivisors(n)): usigma:=n->sum(udivisors(n)[j],j=1..nops(udivisors(n))): uH:=n->n*utau(n)/usigma(n):seq(numer(uH(n)),n=1..81);

ud[n_] := 2^PrimeNu[n]; usigma[n_] := DivisorSum[n, If[GCD[#, n/#] == 1, #, 0]&]; a[1] = 1; a[n_] := Numerator[n*ud[n]/usigma[n]]; Array[a, 100] (* Jean-François Alcover, Dec 03 2016 *)
a[n_] := Numerator[n * Times @@ (2 / (1 + Power @@@ FactorInteger[n]))]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 10 2023 *)

a(n) = {my(f = factor(n)); numerator(n * prod(i=1, #f~, 2/(1 + f[i, 1]^f[i, 2]))); } \\ Amiram Eldar, Mar 10 2023

from sympy import gcd
from sympy.ntheory.factor_ import udivisor_sigma
def A103339(n): return (lambda x, y: y*n//gcd(x,y*n))(udivisor_sigma(n),udivisor_sigma(n,0)) # Chai Wah Wu, Oct 20 2021

a(A006086(n)) = A006087(n). - Reinhard Zumkeller, Mar 17 2012
From Amiram Eldar, Mar 10 2023: (Start)
a(n)/A103340(n) = n*A034444(n)/A034448(n).
a(n)/A103340(n) <= A099377(n)/A099378(n), with equality if and only if n is squarefree (A005117). (End)

A361318 Harmonic means of the infinitary divisors of the infinitary harmonic numbers.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 7, 11, 13, 13, 10, 7, 15, 16, 15, 9, 20, 18, 14, 25, 24, 19, 25, 15, 27, 28, 30, 18, 36, 13, 21, 17, 29, 40, 33, 24, 28, 38, 31, 29, 45, 34, 27, 28, 44, 27, 60, 36, 52, 46, 26, 51, 42, 55, 33, 66, 40, 24, 37, 49, 29, 47, 57, 34, 68, 49, 44

Offset: 1

Amiram Eldar, Mar 09 2023

nonn

Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).

Amiram Eldar, Table of n, a(n) for n = 1..239
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A063947, A361316, A361317.

Similar sequences: A001600, A006087.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; s[1] = 1; s[n_] := n * Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^4], IntegerQ]

ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(ih, ", ")));}

a(n) = A361316(A063947(n)).

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)).

A361784 Harmonic means the bi-unitary divisors of the bi-unitary harmonic numbers (A286325).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 7, 8, 11, 13, 13, 12, 10, 16, 7, 18, 16, 15, 24, 15, 20, 20, 18, 14, 22, 25, 24, 19, 25, 23, 27, 33, 31, 44, 32, 34, 30, 25, 36, 13, 46, 31, 21, 29, 40, 38, 33, 28, 40, 48, 38, 29, 45, 34, 47, 28, 32, 32, 44, 60, 27, 32, 28, 46, 26, 51

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

			a(3) = 3 since A286325(3) = 45, the bi-unitary divisors of 45 are 1, 5, 9, and 45, and their harmonic mean is 3.

Amiram Eldar, Table of n, a(n) for n = 1..313
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A286324, A286325, A361782, A361783.

Similar sequences: A001600, A006087, A361318.

f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^5], IntegerQ]

bhmean(n) = {my(f = factor(n), p, e); n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2)))); }
lista(kmax) = {my(bh); for(k = 1, kmax, bh = bhmean(k); if(denominator(bh) == 1, print1(bh, ", "))); }

a(n) = A361782(A286325(n)).

A361385 a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.

Original entry on oeis.org

0, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 6, 5, 6, 5, 6, 6, 6, 5, 7, 4, 5, 5, 6, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 7, 6, 8, 7, 7, 7, 6, 7, 7, 7, 6, 8, 6, 5, 6, 7, 6, 7, 7, 6, 8, 7, 7, 8, 7, 6, 7, 8, 7, 6, 8, 7, 7, 7, 7, 9, 6, 8, 6, 8, 8, 7

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).

Amiram Eldar, Table of n, a(n) for n = 1..239
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian math. Soc., Vol. 41 (1990), pp. 151-158.

Cf. A006086, A006087, A361384 (analogous unitary sequence).

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ih[1] = 1; ih[n_] := n*Times @@ f @@@ FactorInteger[n]; ic[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); ic[1] = 0; ic /@ Select[Range[10^5], IntegerQ[ih[#]] &]

A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus at A064547
ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(A064547(k), ", ")));}

a(n) = A064547(A063947(n)).

A353038 Unitary harmonic numbers (A006086) that are not unitary arithmetic numbers (A103826).

Original entry on oeis.org

90, 40682250, 81364500, 105773850, 423095400, 1798155450, 14385243600

Offset: 1

Amiram Eldar, Apr 19 2022

There are 290 unitary harmonic numbers below 10^12, and only 7 of them are in this sequence.

			90 is in the sequence since its unitary divisors are {1, 2, 5, 9, 10, 18, 45, 90}, their harmonic mean, 4, is an integer, but their arithmetic mean, 45/2, is not.

The unitary version of A046999.
Subsequence of A006086.
Cf. A006087, A103826.

q[n_] := Module[{f = FactorInteger[n], d, s}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); IntegerQ[n*d/s] && !IntegerQ[s/d]]; Select[Range[5*10^7], q]

cp's OEIS Frontend

A006086 Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).

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A103339 Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

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A361318 Harmonic means of the infinitary divisors of the infinitary harmonic numbers.

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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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A361784 Harmonic means the bi-unitary divisors of the bi-unitary harmonic numbers (A286325).

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A361385 a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.

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A353038 Unitary harmonic numbers (A006086) that are not unitary arithmetic numbers (A103826).

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