A077610 -id:A077610 - cp's OEIS frontend

A380396 a(n) is the sum of the unitary divisors of n that are cubes.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 23 2025

The number of unitary divisors of n that are cubes is A380395(n).

			a(8) = 9 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3, and 1 + 8 = 9.
a(216) = 252 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3, and 1 + 8 + 27 + 216 = 252.

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

Cf. A000578, A046100, A077610, A113061, A358347, A366761, A371334, A380395.

f[p_, e_] := If[Divisible[e, 3], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%3, 1, f[i, 1]^f[i, 2] + 1));}

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * [d is cube], where [] is the Iverson bracket.
a(n) >= 1, with equality if and only if n is not in A366761.
a(n) <= A113061(n), with equality if and only if n is biquadratefree (A046100).
Multiplicative with a(p^e) = p^e + 1 if e is divisible by 3, and 1 otherwise.
Sum_{k=1..n} a(k) ~ c * n^(4/3) / 4, where c = zeta(4/3)/zeta(7/3) = 2.54455250463133711749... .
Dirichlet g.f.: zeta(s) * zeta(3*s-3) / zeta(4*s-3).
In general, the average order of the sum of the unitary divisors that are m-powers is c * n^(1+1/m) / (m+1), where c = zeta(1+1/m)/zeta(2+1/m), and its Dirichlet g.f. is zeta(s) * zeta(m*s-m) / zeta((m+1)*s-m), both for m >= 2.

A383763 The sum of unitary divisors of n that are exponentially squarefree numbers.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68

Offset: 1

Amiram Eldar, May 09 2025

The number of these divisors is A383762(n) and the largest of them is A383764(n).

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

Cf. A005117, A034448, A077610, A209061, A365682, A383762, A383764.

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

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

Multiplicative with a(p^e) = p^e + 1 if e is squarefree (A005117), and 1 otherwise.
a(n) <= A034448(n), with equality if and only if n is an exponentially squarefree number (A209061).
a(n) <= A365682(n), with equality if and only if n is a squarefree number.

A006087 Unitary harmonic means H(n) of the unitary harmonic numbers (A006086).

Original entry on oeis.org

1, 2, 3, 4, 4, 7, 7, 6, 9, 13, 10, 13, 10, 7, 11, 15, 10, 15, 9, 12, 7, 17, 12, 18, 16, 14, 19, 20, 19, 12, 15, 20, 10, 20, 18, 22, 19, 13, 12, 13, 17, 29, 18, 33, 20, 23, 29, 34, 23, 22, 31, 38, 24, 23, 38, 33, 37, 40, 19, 38, 24, 37, 29, 40, 22, 34, 24, 33

Offset: 1

N. J. A. Sloane

Let d(n) and sigma(n) be number and sum of unitary divisors of n; then unitary harmonic mean of unitary divisors is H(n)=n*d(n)/sigma(n).

Each term appears a finite number of times in the sequence (Hagis and Lord, 1975). - Amiram Eldar, Mar 10 2023

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
Peter Hagis, Jr. and Graham Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7.
Peter Hagis, Jr. and Graham Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7. (Annotated scanned copy)

Cf. A006086, A034444, A034448, A077610.

import Data.Ratio ((%), numerator, denominator)
a006087 n = a006087_list !! (n-1)
a006087_list = map numerator $ filter ((== 1) . denominator) $
   map uhm [1..]  where uhm n = (n * a034444 n) % (a034448 n)
-- Reinhard Zumkeller, Mar 17 2012

A034444 := proc(n) 2^nops(ifactors(n)[2]) ; end: A034448 := proc(n) local ans,i,ifs ; ans :=1 ; ifs := ifactors(n)[2] ; for i from 1 to nops(ifs) do ans := ans*(1+ifs[i][1]^ifs[i][2]) ; od ; RETURN(ans) ; end: A006086 := proc(n) n*A034444(n)/A034448(n) ; end: for n from 1 to 5000000 do uhn := A006086(n) : if type(uhn,'integer') then printf("%d, ",uhn) ; fi ; od : # R. J. Mathar, Jun 06 2007

ud[n_] := 2^PrimeNu[n]; usigma[n_] := Sum[ If[ GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := n*ud[n]/usigma[n]; a[1] = 1; Reap[ Do[ If[ IntegerQ[h = a[n]], Print[h]; Sow[h]], {n, 1, 10^7}]][[2, 1]] (* Jean-François Alcover, May 16 2013 *)
uh[n_] := n * Times @@ (2/(1 + Power @@@ FactorInteger[n])); uh[1] = 1; Select[Array[uh, 10^6], IntegerQ] (* Amiram Eldar, Mar 10 2023 *)

{ud(n)=2^omega(n)} {sud(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))} {H(n)=n*ud(n)/sud(n)} for(n=1,10000000,if(((n*ud(n))%sud(n))==0,print1(H(n)","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

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(uh, ", ")));} \\ Amiram Eldar, Mar 10 2023

a(n) = A103339(A006086(n)). - Reinhard Zumkeller, Mar 17 2012

More terms from R. J. Mathar, Jun 06 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 5, 10, 9, 1, 4, 6, 17, 28, 17, 1, 2, 12, 26, 65, 82, 33, 1, 2, 8, 50, 126, 257, 244, 65, 1, 2, 9, 50, 252, 626, 1025, 730, 129, 1, 4, 10, 65, 344, 1394, 3126, 4097, 2188, 257, 1, 2, 18, 82, 513, 2402, 8052, 15626, 16385, 6562, 513, 1, 4, 12, 130, 730, 4097, 16808, 47450, 78126, 65537, 19684, 1025, 1

Offset: 0

Ilya Gutkovskiy, Aug 02 2017

For row r > 0, Sum_{k=1..n} A(r,k) ~ zeta(r+1) * n^(r+1) / ((r+1) * zeta(r+2)). - Vaclav Kotesovec, May 20 2021

			Square array begins:
1,   2,    2,     2,     2,     4,  ...
1,   3,    4,     5,     6,    12,  ...
1,   5,   10,    17,    26,    50,  ...
1,   9,   28,    65,   126,   252,  ...
1,  17,   82,   257,   626,  1394,  ...
1,  33,  244,  1025,  3126,  8052,  ...

Eric Weisstein's World of Mathematics, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Index entries for sequences related to sums of divisors

Rows n=0-8 give: A034444, A034448, A034676, A034677, A034678, A034679, A034680, A034681, A034682.

Cf. A077610, A109974, A285425.

Dirichlet g.f. of row n: zeta(s)*zeta(s-n)/zeta(2*s-n).

A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 9, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 9, 26, 1, 28, 5, 1, 1, 1, 33, 1, 1, 1, 50, 1, 1, 1, 9, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 28, 1, 9, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 90, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset: 1

Amiram Eldar, Feb 18 2023

The number of these divisors is given by A323308.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^(3/2) for n = 1..10^8.
Index entries for sequences related to divisors of numbers.

Cf. A001694, A004709, A034444, A034448, A057521, A077610, A092261, A323308.

Similar sequences: A183097, A360722.

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

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

for(n=1, 100, print1(direuler(p=2, n, (1 - p^3*X^4 - p^2*X^3 + p^3*X^3) / ((1 - X) * (1 - p^2*X^2)))[n], ", ")) \\ Vaclav Kotesovec, Feb 18 2023

Multiplicative with a(p) = 1 and a(p^e) = p^e + 1 for e > 1.
a(n) <= A034448(n), with equality if and only if n is powerful (A001694).
a(n) <= A183097(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - p^(1-s) + p^(2-2*s) - p^(2-3*s)).
From Vaclav Kotesovec, Feb 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 - p^(3-4*s) - p^(2-3*s) + p^(3-3*s)).
Sum_{k=1..n} a(k) ~ c * zeta(3/2) * n^(3/2) / 3, where c = Product_{primes p} (1 + 1/p^(3/2) - 1/p^(5/2) - 1/p^3) = 1.48039182258752809541724060173644... (End)
a(n) = A034448(A057521(n)) (the sum of unitary divisors of the powerful part of n). - Amiram Eldar, Dec 12 2023
a(n) = A034448(n)/A092261(n). - Amiram Eldar, Jun 19 2025

A220218 Numbers where all exponents in its prime factorization are one less than a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Charles R Greathouse IV, Dec 07 2012

nonn

Sequence has positive density, between 0.83 and 0.89; probably about 0.87951.
The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 90, 880, 8796, 87956, 879518, 8795126, 87951173, 879511794, ... The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + Sum_{q prime >= 5} (p-1)/p^q) = 0.87951176583716527413... - Amiram Eldar, Mar 20 2021
Numbers whose sets of unitary divisors (A077610) and modified exponential divisors (A379027) coincide. - Amiram Eldar, Dec 14 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős and Leon Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc., Vol. s3-2, No. 1 (1952), pp. 257-271; alternative link.

Apart from the first term, a subsequence of A096432.

Cf. A010051, A124010, A077610, A379027.

a220218 n = a220218_list !! (n-1)
a220218_list = 1 : filter
               (all (== 1) . map (a010051' . (+ 1)) . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Select[Range[100],AllTrue[Transpose[FactorInteger[#]][[2]]+1,PrimeQ]&] (* Harvey P. Dale, Sep 29 2014 *)

is(n)=vecmin(apply(n->isprime(n+1),factor(max(n,2))[,2])) \\ Charles R Greathouse IV, Dec 07 2012

A222084 Number of the least divisors of n whose LCM is equal to n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 5, 2, 4, 3, 3, 2, 6, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 6, 2, 3, 3, 5, 2, 5, 2, 4, 4, 3, 2, 8, 3, 5, 3, 4, 2, 7, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 3, 5, 2, 4, 3, 4, 2, 7, 2, 3, 5, 4, 3, 5, 2, 7, 5, 3, 2, 6, 3, 3, 3

Offset: 1

Paolo P. Lava, Feb 07 2013

nonn

If we write n as the product of its prime factors, n = p1^a1*p2^a2*p3^a3*...*pr^ar, then tau#(n) gives the number of divisors from 1 to max(p1^a1, p2^a2, p3^a3, ..., pr^ar).
In general tau#(n) <= tau(n).
Also, tau#(n) = tau(n) is A000961, tau#(n) < tau(n) is A024619.
For any prime number p tau(p) = tau#(p) = 2.
tau#(n) = 3 only for semiprimes (A001358).

			For n=40, the divisors are (1, 2, 4, 5, 8, 10, 20, 40), so tau(40)=8.
lcm(1, 2, 4, 5, 8) = 40, but lcm(1, 2, 4, 5) = 20 < 40, so tau#(40)=5.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222085.

with(numtheory);
A222084:=proc(q)
local a,b,c,j,n; print(1);
for n from 2 to q do a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
  for j from 1 to b do if a[j]=c then break; fi; od; print(j); od; end:
A222084(100000);

Table[Count[ Divisors[n] , q_Integer /; q <= Max[Power @@@ FactorInteger[n]]], {n, 87}] (* Wouter Meeussen, Feb 09 2013 *)

a(n) = {my(d = divisors(n), k = 1); while (lcm(vector(k, j, d[j])) != n, k++); k;} \\ Michel Marcus, Mar 13 2018

A225174 Square array read by antidiagonals: T(m,n) = greatest common unitary divisor of m and n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1

Offset: 1

N. J. A. Sloane, May 01 2013

			Array begins
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, ...
1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, ...
1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, ...
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, ...
1, 2, 3, 1, 1, 6, 1, 1, 1, 2, 1, 3, ...
1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, ...
...
The unitary divisors of 3 are 1 and 3, those of 6 are 1,2,3,6; so T(6,3) = T(3,6) = 3.

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Antti Karttunen, Table of n, a(n) for n = 1..20100; the first 200 antidiagonals of the array

See A034444, A077610 for unitary divisors of n.

Different from A059895.

# returns the greatest common unitary divisor of m and n
f:=proc(m,n)
local i,ans;
ans:=1;
for i from 1 to min(m,n) do
if ((m mod i) = 0) and (igcd(i,m/i) = 1)  then
   if ((n mod i) = 0) and (igcd(i,n/i) = 1)  then ans:=i; fi;
fi;
od;
ans; end;

f[m_, n_] := Module[{i, ans=1}, For[i=1, i<=Min[m, n], i++, If[Mod[m, i]==0 && GCD[i, m/i]==1, If[Mod[n, i]==0 && GCD[i, n/i]==1, ans=i]]]; ans];
Table[f[m-n+1, n], {m, 1, 14}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 19 2018, translated from Maple *)

up_to = 20100; \\ = binomial(200+1,2)
A225174sq(m,n) = { my(a=min(m,n),b=max(m,n),md=0); fordiv(a,d,if(0==(b%d)&&1==gcd(d,a/d)&&1==gcd(d,b/d),md=d)); (md); };
A225174list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A225174sq((a-(col-1)),col))); (v); };
v225174 = A225174list(up_to);
A225174(n) = v225174[n]; \\ Antti Karttunen, Nov 28 2018

T(m,n) = T(n,m) = A165430(n,m).

A366536 The number of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 12 2023

The number of unitary divisors of the squarefree numbers (A005117) is the same as the number of divisors of the squarefree numbers (A072048), because all the divisors of a squarefree number are unitary.

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

Cf. A004709, A005117, A034444, A072048, A077610, A358040, A366440, A366537.

Similar sequences: A366534, A366538.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

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

from sympy.ntheory.factor_ import udivisor_count
from sympy import mobius, integer_nthroot
def A366536(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return udivisor_count(m) # Chai Wah Wu, Aug 05 2024

a(n) = A034444(A004709(n)).

A366537 The sum of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 18, 30, 20, 30, 32, 36, 24, 26, 42, 40, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104, 100, 96

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A002117, A004709, A005117, A034448, A062822, A077610, A358040, A366440, A366536.

Similar sequences: A034676, A366535, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # < 3 &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

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

from sympy.ntheory.factor_ import udivisor_sigma
from sympy import mobius, integer_nthroot
def A366537(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return udivisor_sigma(m) # Chai Wah Wu, Aug 05 2024

a(n) = A034448(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 1/p^4 - 1/p^5) = 1.665430860774244601005... .
The asymptotic mean of the unitary abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = c / zeta(3) = 1.38548421160152785073... .

cp's OEIS Frontend

A380396 a(n) is the sum of the unitary divisors of n that are cubes.

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A383763 The sum of unitary divisors of n that are exponentially squarefree numbers.

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A006087 Unitary harmonic means H(n) of the unitary harmonic numbers (A006086).

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A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

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A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).

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A220218 Numbers where all exponents in its prime factorization are one less than a prime.

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A222084 Number of the least divisors of n whose LCM is equal to n.

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