A103340 -id:A103340 - cp's OEIS frontend

A348654 Indices of records in the sequence of unitary harmonic means A103339(k)/A103340(k).

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 28, 30, 42, 60, 84, 105, 120, 140, 180, 210, 330, 390, 420, 660, 780, 840, 1092, 1155, 1260, 1540, 1820, 1980, 2310, 2730, 3570, 3990, 4290, 4620, 5460, 7140, 7980, 8580, 9240, 10920, 13860, 16380, 20020, 23940, 25740, 27720, 30030

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

			The unitary harmonic means of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. The next record, A103339(10)/A103340(10) = 20/9, occurs at 10. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 10.

Amiram Eldar, Table of n, a(n) for n = 1..170

The unitary version of A179971.

Cf. A103339, A103340.

f[p_, e_] := 2/(1 + p^(-e)); uhmeam[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; max = 0; Do[u1 = uhmeam[n]; If[u1 > max, max = u1; AppendTo[s, n]], {n, 1, 10^5}]; s

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

Original entry on oeis.org

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)

A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 128, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 256, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 09 2023

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).

Similar sequences: A099377, A103339.

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}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(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)))); }

a(n) = numerator(n*A037445(n)/A049417(n)).
a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.
a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.

A361317 Denominators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 15, 5, 9, 6, 5, 7, 3, 2, 17, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 51, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 17, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 85, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57

Offset: 1

Amiram Eldar, Mar 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A037445, A049417, A077609, A063947 (positions of 1's), A361316 (numerators).

Similar sequences: A099378, A103340.

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}]]; a[1] = 1; a[n_] := Denominator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); denominator(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)))); }

a(n) = denominator(n*A037445(n)/A049417(n)).

A361783 Denominators of the harmonic means of the bi-unitary divisors of the positive integers.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 15, 5, 9, 6, 5, 7, 3, 2, 27, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 21, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 9, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 119, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57

Offset: 1

Amiram Eldar, Mar 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A222266, A286324, A286325 (positions of 1's), A361782 (numerators).

Similar sequences: A099378, A103340, A361317.

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))]; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); denominator(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))))); }

a(n) = denominator(n*A286324(n)/A188999(n)).

A353692 a(n) is the least number k > n such that uh(k)/uh(n) is an integer, where uh(n) is the harmonic mean of the unitary divisors of n, or -1 if no such k exists.

Original entry on oeis.org

6, 20, 45, 72, 30, 60, 42, 272, 756, 120, 66, 18, 78, 140, 1890, 720, 102, 180, 114, 24, 315, 220, 138, 360, 150, 260, 3321, 504, 174, 7560, 186, 1440, 495, 340, 210, 52416, 222, 380, 585, 1360, 246, 420, 258, 792, 1512, 460, 282, 720, 294, 600, 765, 936, 318

Offset: 1

Amiram Eldar, May 04 2022

nonn

			a(2) = 20 since 20 is the least number > 2 such that uh(20)/uh(2) = (8/3)/(4/3) = 2 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..399

Cf. A103339, A103340.

Similar sequences: A069789, A069797, A069805, A353691.

uh[n_] := Module[{f = FactorInteger[n]}, n*2^Length[f]/Times @@ (1 + Power @@@ f)]; a[n_] := Module[{k = n + 1, uhn = uh[n]}, While[!IntegerQ[uh[k]/uhn], k++]; k]; Array[a, 30]

a(p) = 6*p for a prime p > 3.

A348656 Numbers with record values of the denominator of the harmonic mean of their unitary divisors.

Original entry on oeis.org

1, 2, 4, 8, 16, 26, 32, 50, 58, 64, 98, 106, 122, 128, 178, 194, 202, 218, 226, 242, 250, 256, 346, 362, 386, 394, 458, 466, 482, 512, 698, 706, 722, 746, 778, 794, 802, 818, 842, 866, 898, 914, 922, 1018, 1024, 1402, 1418, 1466, 1514, 1522, 1538, 1546, 1594, 1618

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

The corresponding record values are 1, 3, 5, 9, 17, 21, 33, 39, 45, 65, 75, ... (see the link for more values).

			The first 8 terms of A103340 are 1, 3, 2, 5, 3, 1, 4 and 9. The record values, 1, 3, 5 and 9, occur at 1, 2, 4 and 8, the first 4 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Table of n, a(n), A103340(a(n)) for n = 1..10000

The unitary version of A348414.

Cf. A103339, A103340.

f[p_, e_] := 2/(1 + p^(-e)); d[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; dm = 0; s = {}; Do[dn = d[n]; If[dn > dm, dm = dn; AppendTo[s, n]], {n, 1, 1000}]; s

A348657 Numbers k such that k and k+1 have the same denominator of the harmonic means of their unitary divisors.

Original entry on oeis.org

266, 321, 1015, 2544, 4004, 4277, 5016, 15861, 28461, 47613, 63546, 135078, 137333, 203709, 207024, 265489, 344217, 383466, 517610, 603687, 787156, 798625, 876469, 1100835, 1713865, 2062863, 2246923, 2349390, 2666741, 3013830, 3961129, 5048409, 6148960, 6491717

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

Numbers k such that A103340(k) = A103340(k+1).
The common denominators of k and k+1 are 30, 36, 36, 153, 15, 96, 45, 936, ...
Can 3 consecutive numbers have the same denominator of harmonic mean of unitary divisors? There are no such numbers below 2.5*10^10.

			266 is a term since the harmonic means of the unitary divisors of 266 and 267 are 133/30 and 89/30, respectively, and both have the denominator 30.

Amiram Eldar, Table of n, a(n) for n = 1..400

The unitary version of A348415.

Cf. A103339, A103340.

f[p_, e_] := 2/(1 + p^(-e)); d[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Select[Range[10^5], d[#] == d[# + 1] &]

cp's OEIS Frontend

A348654 Indices of records in the sequence of unitary harmonic means A103339(k)/A103340(k).

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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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A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

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A361317 Denominators of the harmonic means of the infinitary divisors of the positive integers.

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A361783 Denominators of the harmonic means of the bi-unitary divisors of the positive integers.

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A353692 a(n) is the least number k > n such that uh(k)/uh(n) is an integer, where uh(n) is the harmonic mean of the unitary divisors of n, or -1 if no such k exists.

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