A077610 -id:A077610 - cp's OEIS frontend

A358346 a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

1, 3, 4, 1, 6, 12, 8, 9, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 36, 1, 42, 28, 8, 30, 72, 32, 33, 48, 54, 48, 1, 38, 60, 56, 54, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 84, 72, 72, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Amiram Eldar, Nov 11 2022

The number of unitary divisors of n that are exponentially odd is A055076(n).

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

Cf. A000290, A005117, A055076, A077610, A268335, A351569.

Similar sequences: A033634, A034448, A035316, A358347.

f[p_, e_] := 1 + If[OddQ[e], p^e, 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i,2]%2,  f[i,1]^f[i,2], 0));}

a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A033634(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^e + 1 if e is odd, and 1 otherwise.
a(n) = A034448(n)/A358347(n).
Sum_{k=1..n} a(k) ~ n^2/2.
From Amiram Eldar, Sep 14 2023: (Start)
a(n) = A034448(A350389(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(2*s-1)). (End)

A034676 Sum of squares of unitary divisors of n.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 65, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 650, 626, 850, 730, 850, 842, 1300, 962, 1025, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130

Offset: 1

Erich Friedman

Also sum of unitary divisors of n^2. - Vladeta Jovovic, Nov 13 2001

If b(n,k)=sum of k-th powers of unitary divisors of n then b(n,k) is multiplicative with b(p^e,k)=p^(k*e)+1. - Vladeta Jovovic, Nov 13 2001

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.

Cf. A034444, A034448, A034677, A034678-A034682.

Cf. A077610.

a034676 = sum . map (^ 2) . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

A034676 := proc(n)
    a :=1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        a := a*(p^(2*e)+1) ;
    end do:
    a ;
end proc:
seq(A034676(n),n=1..40) ; # R. J. Mathar, Jul 12 2024

f[n_] := Block[{d = Divisors@ n}, Plus @@ (Select[d, GCD[#, n/#] == 1 &]^2)]; Array[f, 50] (* Robert G. Wilson v, Mar 04 2011 *)
f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

A034676_vec(len)={
        a000012=direuler(p=2,len, 1/(1-X)) ;
        a000290=direuler(p=2,len, 1/(1-p^2*X)) ;
        a000290x=direuler(p=2,len, 1-p^2*X^2) ;
        dirmul(dirmul(a000012,a000290),a000290x)
}
A034676_vec(70) ; /* via D.g.f., R. J. Mathar, Mar 05 2011 */

Multiplicative with a(p^e)=p^(2*e)+1.
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2*s-2). - R. J. Mathar, Mar 04 2011
Sum_{k=1..n} a(k) ~ 30 * Zeta(3) * n^3 / Pi^4. - Vaclav Kotesovec, Jan 11 2019
Sum_{k>=1} 1/a(k) = 1.5594563610641446770272272038182777336348840179730233519185104374159616326... - Vaclav Kotesovec, Sep 20 2020

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)

A103340 Denominator 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, 3, 2, 5, 3, 1, 4, 9, 5, 9, 6, 5, 7, 3, 2, 17, 9, 5, 10, 3, 8, 9, 12, 3, 13, 21, 14, 5, 15, 3, 16, 33, 4, 27, 12, 25, 19, 15, 14, 27, 21, 2, 22, 15, 1, 9, 24, 17, 25, 39, 6, 35, 27, 7, 18, 9, 20, 45, 30, 1, 31, 12, 20, 65, 21, 3, 34, 45, 8, 9, 36, 5, 37, 57, 26, 25, 24, 7, 40, 51, 41, 63

Offset: 1

Emeric Deutsch, Jan 31 2005

			1, 4/3, 3/2, 8/5, 5/3, 2, ...
a(8) = 9 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. A103339 (numerators), A099377, A099378.

Cf. A034444, A034448, A077610.

import Data.Ratio ((%), denominator)
a103340 = denominator . 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(denom(uH(n)),n=1..90);

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

a(n) = {my(f = factor(n)); denominator(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 A103340(n): return (lambda x, y: x//gcd(x,y*n))(udivisor_sigma(n),udivisor_sigma(n,0)) # Chai Wah Wu, Oct 20 2021

a(A006086(n)) = 1. - Reinhard Zumkeller, Mar 17 2012

A192066 Sum of the odd unitary divisors of n.

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 8, 1, 10, 6, 12, 4, 14, 8, 24, 1, 18, 10, 20, 6, 32, 12, 24, 4, 26, 14, 28, 8, 30, 24, 32, 1, 48, 18, 48, 10, 38, 20, 56, 6, 42, 32, 44, 12, 60, 24, 48, 4, 50, 26, 72, 14, 54, 28, 72, 8, 80, 30, 60, 24, 62, 32, 80, 1, 84, 48, 68, 18, 96, 48, 72, 10, 74, 38, 104, 20, 96, 56, 80, 6

Offset: 1

R. J. Mathar, Jun 22 2011

The unitary analog of A000593.

			n=9 has the divisors 1, 3 and 9, of which 3 is not a unitary divisor because gcd(3,9/3) = gcd(3,3) != 1. This leaves 1 and 9 as unitary divisors which sum to a(9) = 1+9 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, section 4.2.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A068068, A034448.

Cf. A077610, A206787.

a192066 = sum . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

unitaryOddSigma := proc(n,k) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if type(d,'odd') then if igcd(d,n/d) = 1 then a := a+d^k ; end if; end if; end do: a ; end proc:
A := proc(n) unitaryOddSigma(n,1) ;end proc:

a[n_] := DivisorSum[n, Boole[OddQ[#] && GCD[#, n/#] == 1]*#&];
Array[a, 80] (* Jean-François Alcover, Nov 16 2017 *)
f[2, p_] := 1; f[p_, e_] := p^e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, if ((gcd(d, n/d)==1) && (d%2), d)); \\ Michel Marcus, Nov 17 2017

a(n) = Sum_{d|n, d odd, gcd(d,n/d)=1} d.
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s))/( zeta(2s-1)*(1-2^(1-2s)) ).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (21*zeta(3)). - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = 1, and a(p^e) = p^e + 1 for p > 2. - Amiram Eldar, Sep 18 2020

A066874 Number of partitions of n into unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 8, 2, 2, 2, 11, 2, 12, 2, 14, 14, 2, 2, 17, 2, 17, 18, 20, 2, 20, 2, 23, 2, 22, 2, 742, 2, 2, 26, 29, 26, 27, 2, 32, 30, 29, 2, 1654, 2, 32, 32, 38, 2, 36, 2, 41, 38, 37, 2, 44, 38, 38, 42, 47, 2, 3004, 2, 50, 42, 2, 44, 5257, 2, 47, 50, 5066, 2, 47, 2, 59, 54, 52, 50

Offset: 1

Naohiro Nomoto, Jan 26 2002

			a(12) = 12 because the unitary divisors of 12 are 1, 3, 4 and 12; and the partitions are 12, 4+4+4, 4+4+3+1, 4+4+(4x1), 4+3+3+1+1, 4+3+(5x1), 4+(8x1), 3+3+3+3, 3+3+3+1+1+1, 3+3+(6x1), 3+(9x1) and 12x1.

David A. Corneth, Table of n, a(n) for n = 1..10001 (first 329 terms by Antti Karttunen)
David A. Corneth, PARI program

Cf. A018818, A034444, A077610, A285614, A286852.

unitary_divisors(n) = select(d -> (1==gcd(d,n/d)), divisors(n));
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A066874(n) = partitions_into(n,vecsort(unitary_divisors(n), , 4)); \\ Antti Karttunen, Aug 06 2018

See Corneth link. \\ David A. Corneth, Aug 12 2018

More terms from David Wasserman, Nov 21 2002

A115063 Natural numbers of the form p^F(n_p)q^F(n_q)r^F(n_r)...z^F(n_z), where p,q,r,... are distinct primes and F(n) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67

Offset: 1

Giovanni Teofilatto, Mar 01 2006

The complementary sequence is 16, 48, 64, 80, 81, 112, 128, 144, 162, 176, 192, 208, 240, 272, 304, 320, 324, 336, 368, 384, 400, ... - R. J. Mathar, Apr 22 2010
Or exponentially Fibonacci numbers. - Vladimir Shevelev, Nov 15 2015
Sequences A004709, A005117, A046100 are subsequences. - Vladimir Shevelev, Nov 16 2015
Let h_k be the density of the subsequence of A115063 of numbers whose prime power factorization has the form Product_i p_i^e_i where the e_i all squares <= k^2. Then for every k>1 there exists eps_k>0 such that for any x from the interval (h_k-eps_k, h_k) there is no sequence S of positive integers such that x is the density of numbers whose prime power factorization has the form Product_i p_i^e_i where the e_i are all in S. For a proof, see [Shevelev], the second link. - Vladimir Shevelev, Nov 17 2015
Numbers whose sets of unitary divisors (A077610) and Zeckendorf-infinitary divisors (see A318465) coincide. Also, numbers whose sets of unitary divisors and dual-Zeckendorf-infinitary divisors (see A331109) coincide. - Amiram Eldar, Aug 09 2024

			12 is a term, since 12=2^2*3^1 and the exponents 2 and 1 are terms of Fibonacci sequence (A000045). - _Vladimir Shevelev_, Nov 15 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), 385-395.

Cf. A004709, A005117, A046100, A077610, A115975, A197680, A209061, A318465, A331109.

fibQ[n_] := IntegerQ @ Sqrt[5 n^2 - 4] || IntegerQ @ Sqrt[5 n^2 + 4]; aQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], fibQ]; Select[Range[100], aQ] (* Amiram Eldar, Oct 06 2019 *)

Sum_{i<=x, i is in A115063} 1 = h*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and h = Product_{prime p}(1 + Sum_{i>=2} (u(i)-u(i-1))/p^i) = 0.944335905... where u(n) is the characteristic function of sequence A000045. The calculations of h over the formula were done independently by Juan Arias-de-Reyna and Peter J. C. Moses.

For a proof of the formula, see [Shevelev], the first link. - Vladimir Shevelev, Nov 17 2015

a(35) inserted by Amiram Eldar, Oct 06 2019

A229996 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m. The sequence (a(n)) consists of successive numbers m which d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) is an integer.

Original entry on oeis.org

1, 10, 65, 130, 260, 340, 1105, 1972, 2210, 4420, 8840, 9860, 15650, 20737, 32045, 41474, 44200, 51272, 55250, 64090, 75140, 82948, 103685, 128180, 207370, 207553, 221000, 256360, 352529, 414740, 415106, 512720, 532100, 705058, 759025, 813800, 829480, 830212

Offset: 1

Clark Kimberling, Oct 31 2013

The integer sums d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) are given by A229999. - Clark Kimberling, Jun 16 2018

Also numbers m such that the sum of the squares of the unitary divisors of m is divisible by m (the unitary version of A046762). - Amiram Eldar, Jun 16 2018

			The first 10 sums: 1, 5/2, 10/3, 17/4, 26/5, 25/3, 50/7, 65/8, 82/9, 13, so that a(1) = 1 and a(10) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..333 (terms below 10^10)

Cf. A034676, A046762, A229994, A077610, A229999.

z = 1000; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[Plus @@ t[n], {n, 1, z}]; a[n_] := If[IntegerQ[s[[n]]], 1, 0]; u = Table[a[n], {n, 1, z}]; Flatten[Position[u, 1]]  (* A229996 *)
usigma2[n_] :=  If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n]^2)]; seqQ[n_] := Divisible[usigma2[n], n]; Select[Range[10^6], seqQ] (* Amiram Eldar, Jun 16 2018 *)

is(n) = {my(f = factor(n)); !(prod(i = 1, #f~, f[i,1]^(2*f[i,2]) + 1) % n);} \\ Amiram Eldar, Jun 16 2024

Definition corrected by Clark Kimberling, Jun 16 2018

A366534 The number of unitary divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 2, 8, 2, 4, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 8, 4

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A034444, A077610, A268335, A366438, A366535.

Similar sequences: A366536, A366538.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(2^(#e), ", ")));

a(n) = A034444(A268335(n)).

A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 9, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 36, 42, 28, 30, 72, 32, 33, 48, 54, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 54, 84, 72, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 108, 90, 112

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A034448, A065463, A077610, A268335, A366439, A366534.

Similar sequences: A034676, A366537, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, OddQ], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

a(n) = A034448(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 1.92835521961603199612..., d = A065463 is the asymptotic density of the exponentially odd numbers.
The asymptotic mean of the unitary abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = c * d = 1.35841479521454692063... .

cp's OEIS Frontend

A358346 a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

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A034676 Sum of squares of unitary divisors of n.

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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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A103340 Denominator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

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A192066 Sum of the odd unitary divisors of n.

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A066874 Number of partitions of n into unitary divisors of n.

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A115063 Natural numbers of the form p^F(n_p)*q^F(n_q)*r^F(n_r)*...*z^F(n_z), where p,q,r,... are distinct primes and F(n) is a Fibonacci number.

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A115063 Natural numbers of the form p^F(n_p)q^F(n_q)r^F(n_r)...z^F(n_z), where p,q,r,... are distinct primes and F(n) is a Fibonacci number.