A222266 -id:A222266 - cp's OEIS frontend

A004709 Cubefree numbers: numbers that are not divisible by any cube > 1.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 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, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Steven Finch, Jun 14 1998

Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - Michael Somos, Sep 22 2005
The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = A088453. - Gerard P. Michon, May 06 2009
The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - Amiram Eldar, Mar 12 2021
From Amiram Eldar, Feb 26 2024: (Start)
Numbers whose sets of unitary divisors (A077610) and bi-unitary divisors (A222266) coincide.
Number whose all divisors are (1+e)-divisors, or equivalently, numbers k such that A049599(k) = A000005(k). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Gérard P. Michon, On the number of cubefree integers not exceeding N.
Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint, arXiv:1511.03860 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cubefree.

Complement of A046099.
Cf. A005117 (squarefree), A067259 (cubefree but not squarefree), A046099 (cubeful).
Cf. A160112, A160113, A160114 & A160115: On the number of cubefree integers. - Gerard P. Michon, May 06 2009
Cf. A030078.
Cf. A001221, A124010, A212793.
Cf. A000005, A049599, A077610, A222266, A376365, A376366.

a004709 n = a004709_list !! (n-1)
a004709_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

isA004709 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) > 2 then
            return false;
        end if;
    end do:
    true ;
end proc:

Select[Range[6!], FreeQ[FactorInteger[#], {, k /; k > 2}] &] (* Jan Mangaldan, May 07 2014 *)

{a(n)= local(m,c); if(n<2, n==1, c=1; m=1; while( cvecmax(factor(m)[,2]), c++)); m)} /* Michael Somos, Sep 22 2005 */

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) == n
print(list(filter(ok, range(1, 86)))) # Michael S. Branicky, Aug 16 2021

from sympy import mobius, integer_nthroot
def A004709(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 m # Chai Wah Wu, Aug 05 2024

A066990(a(n)) = a(n). - Reinhard Zumkeller, Jun 25 2009
A212793(a(n)) = 1. - Reinhard Zumkeller, May 27 2012
A124010(a(n),k) <= 2 for all k = 1..A001221(a(n)). - Reinhard Zumkeller, Mar 04 2015
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Dec 27 2022

A077609 Triangle in which n-th row lists infinitary divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 4, 7, 28, 1, 29, 1

Offset: 1

Eric W. Weisstein, Nov 11 2002

The first difference from the triangle A222266 (bi-unitary divisors of n) is in row n = 16; indeed, the 16th row of A222266 is (1, 2, 8, 16) while the 16th of this sequence here is (1, 16). - Bernard Schott, Mar 10 2023

The concept of infinitary divisors was introduced by Cohen (1990). - Amiram Eldar, Mar 09 2024

			The first few rows are:
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;
  1, 13;
  1, 2, 7, 14;
  1, 3, 5, 15;
  1, 16;
  1, 17;

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (2) (1993) 373-384.
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A027750, A037445 (row lengths), A049417 (row sums).

Cf. A222266.

import Data.List ((\\))
a077609 n k = a077609_row n !! (k-1)
a077609_row n = filter
   (\d -> d == 1 || null (a213925_row d \\ a213925_row n)) $ a027750_row n
a077609_tabf = map a077609_row [1..]
-- Reinhard Zumkeller, Jul 10 2013

# see the function idivisors() in A049417. # R. J. Mathar, Oct 05 2017

f[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ; Array[f, 30] // Flatten (* Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 29 2005 *) (* edited by Michael De Vlieger, Jun 07 2016 *)

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k,2]); bde = binary(valuation(d, f[k,1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)););); return (1);}
row(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k]));); idiv;} \\ Michel Marcus, Feb 15 2016

A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 27, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 63, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 108, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 119, 84, 144, 68, 90, 96, 144, 72, 150, 74, 114, 104, 100

Offset: 1

R. J. Mathar, Apr 15 2011

The sequence of bi-unitary perfect numbers obeying a(n) = 2*n consists of only 6, 60, 90 [Wall].

Row sum of row n of the irregular table of the bi-unitary divisors, A222266.

			The divisors of n=16 are d=1, 2, 4, 8 and 16. The greatest common unitary divisor of (1,16) is 1, of (2,8) is 1, of (4,4) is 4, of (8,2) is 1, of (16,1) is 1 (see A165430). So 1, 2, 8 and 16 are bi-unitary divisors of 16, which sum to a(16) = 1 + 2 + 8 + 16 = 27.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, J. Austral. Math. Soc. 23 (series A) (1977), 9-27.
József Sándor and Borislav Crstici, Perfect numbers: Old and new issues; perspectives, in Handbook of number theory, II, p. 45.
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009), Article 09.5.2.
Charles R. Wall, Bi-unitary perfect numbers, Proc. Am. Math. Soc. 33 (1) (1972), 39-42.
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017.

Cf. A222266, A027748, A124010, A034448, A319072.

a188999 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = (p ^ (e + 1) - 1) `div` (p - 1) - (1 - m) * p ^ e' where
           (e', m) = divMod e 2
-- Reinhard Zumkeller, Mar 04 2013

A188999 := proc(n) local a,e,p,f; a :=1 ; for f in ifactors(n)[2] do e := op(2,f) ; p := op(1,f) ; if type(e,'odd') then a := a*(p^(e+1)-1)/(p-1) ; else a := a*((p^(e+1)-1)/(p-1)-p^(e/2)) ; end if; end do: a ; end proc:
seq( A188999(n),n=1..80) ;

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, # &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 76}] (* Michael De Vlieger, May 07 2017 *)
a[n_] := If[n==1, 1, Product[{p, e} = pe; If[OddQ[e], (p^(e+1)-1)/(p-1), ((p^(e+1)-1)/(p-1)-p^(e/2))], {pe, FactorInteger[n]}]]; Array[a, 80] (* Jean-François Alcover, Sep 22 2018 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = vecsum(biudivs(n)); \\ Michel Marcus, May 07 2017

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; e = f[i,2]; f[i,1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 09 2017

from math import prod
from sympy import factorint
def A188999(n): return prod((p**(e+1)-1)//(p-1)-(0 if e&1 else p**(e>>1)) for p,e in factorint(n).items()) # Chai Wah Wu, Dec 28 2024

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd, a(p^e) = (p^(e+1)-1)/(p-1) -p^(e/2) if e is even.
a(n) = A000203(n) - A319072(n). - Omar E. Pol, Sep 29 2018
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 2/p^(2*s-1) + 1/p^(3*s-2) + 1/p^(3*s-1) - 1/p^(4*s-2)). - Amiram Eldar, Aug 28 2023

A286324 a(n) is the number of bi-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 6, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8

Offset: 1

Michel Marcus, May 07 2017

a(n) is the number of terms of the n-th row of A222266.

			From _Michael De Vlieger_, May 07 2017: (Start)
a(1) = 1 since 1 is the empty product; all divisors of 1 (i.e., 1) have a greatest common unitary divisor that is 1. 1 is a unitary divisor of all numbers n.
a(p) = 2 since 1 and p have greatest common unitary divisor 1.
a(6) = 4 since the divisor pairs {1, 6} and {2, 3} have greatest common unitary divisor 1.
a(24) = 8 since {1, 24}, {2, 12}, {3, 8}, {4, 6} have greatest unitary divisors {1, {1, 3, 8, 24}}, {{1, 2}, {1, 3, 4, 12}}, {{1, 3}, {1, 8}}, {{1, 4}, {1, 2, 3, 6}}: 1 is the greatest common unitary divisor among all 4 pairs. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
D. Suryanarayana, The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions pp 273-282, Lecture Notes in Mathematics book series (LNM, volume 251).
Eric Weisstein's World of Mathematics, Biunitary Divisor.

Cf. A222266, A188999, A293185 (indices of records), A340232, A350390.

Cf. A000005, A034444 (unitary), A037445 (infinitary).

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, 1 &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 90}] (* Michael De Vlieger, May 07 2017 *)
f[p_, e_] := If[OddQ[e], e + 1, e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120] (* Amiram Eldar, Dec 19 2018 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = #biudivs(n);

a(n)={my(f=factor(n)[,2]); prod(i=1, #f, my(e=f[i]); e + e % 2)} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (X^3 - X^2 + X + 1) / ((X-1)^2 * (X+1)))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = e + (e mod 2). - Andrew Howroyd, Aug 05 2018
a(A340232(n)) = 2*n. - Bernard Schott, Mar 12 2023
a(n) = A000005(A350390(n)) (the number of divisors of the largest exponentially odd number dividing n). - Amiram Eldar, Sep 01 2023
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Let f(s) = Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (p-1)/((p+1)*p^2)) = A306071 = 0.80733082163620503914865427993003113402584582508155664401800520770441381...,
f'(1) = f(1) * Sum_{p prime} 2*(p^2 - p - 1) * log(p) /(p^4 + 2*p^3 + 1) = f(1) * 0.40523703144422392508596509911218523410441417240419849262346362977537989... = f(1) * A306072
and gamma is the Euler-Mascheroni constant A001620. (End)

A322791 Irregular triangle read by rows in which the n-th row lists the exponential divisors (or e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 12, 48, 7, 49

Offset: 1

Amiram Eldar, Dec 26 2018

			The table starts
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13, No. 2 (2010), Article 10.3.7.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471, alternative link.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A049419 (row lengths), A051377 (row sums).

Cf. A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary).

A322791 := proc(n)
    local expundivs ,d,isue,p,ai,bi;
    expudvs := {} ;
    for d in numtheory[divisors](n) do
        isue := true ;
        for p in numtheory[factorset](n) do
            ai := padic[ordp](n,p) ;
            bi := padic[ordp](d,p) ;
            if bi > 0 then
                if modp(ai,bi) <>0 then
                    isue := false;
                end if;
            else
                isue := false ;
            end if;
        end do;
        if isue then
            expudvs := expudvs union {d} ;
        end if;
    end do:
    sort(expudvs) ;
end proc:
seq(op(A322791(n)),n=1..40) ; # R. J. Mathar, Mar 06 2023

divQ[n_, m_] := (n > 0 && m>0 && Divisible[n, m]); expDivQ[n_, d_] := Module[ {f=FactorInteger[n]}, And@@MapThread[divQ, {f[[;; , 2]], IntegerExponent[ d, f[[;; , 1]]]} ]]; expDivs[1]={1}; expDivs[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d, expDivQ[n, #]&] ]; Table[expDivs[n], {n, 1, 50}] // Flatten

isexpdiv(f, d) = { my(e); for (i=1, #f~, e = valuation(d, f[i, 1]); if(!e || (e && f[i, 2] % e), return(0))); 1; }
row(n) = {my(d = divisors(n), f = factor(n), ediv = []); if(n == 1, return([1])); for(i=2, #d, if(isexpdiv(f, d[i]), ediv = concat(ediv, d[i]))); ediv; } \\ Amiram Eldar, Mar 27 2023

A286325 Bi-unitary harmonic numbers.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 672, 2970, 5460, 8190, 9072, 9100, 10080, 15925, 22680, 22848, 27300, 30240, 40950, 45360, 54600, 81900, 95550, 99792, 136500, 163800, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 409500, 472500, 491400

Offset: 1

Michel Marcus, May 07 2017

nonn

A number m is a term if the sum of its bi-unitary divisors, A188999(m) divides the product of m by the number of its bi-unitary divisors A286324(m).

Numbers k whose harmonic mean of their bi-unitary divisors, A361782(k)/A361783(k), is an integer. - Amiram Eldar, Mar 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..313 (all terms below 10^10, first 40 terms from Michel Marcus)
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011. See pp. 13-20, but beware of typos and possible errors.

Cf. A222266, A188999, A286324, A361782, A361783, A361784.

Cf. A001599 (Ore harmonic), A006086 (unitary harmonic).

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))]; bhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; bhQ[1] = True; Select[Range[10^5], bhQ] (* Amiram Eldar, Mar 24 2023 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
isok(n) = my(v=biudivs(n)); denominator(n*#v/vecsum(v))==1;

A362852 The number of divisors of n that are both bi-unitary and exponential.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 05 2023

First differs from A061704 at n = 128, and from A304327 and abs(A307428) at n = 64.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a bi-unitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e but k != e/2.
The least term that is higher than 2 is a(64) = 3.
This sequence is unbounded. E.g., a(2^(2^prime(n))) = prime(n).

			a(8) = 2 since 8 has 2 divisors that are both bi-unitary and exponential: 2 and 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A000005, A004709, A049419, A188999, A222266, A286324, A322791, A359411, A362853 (indices of records), A362854.

Cf. A061704, A304327, A307428.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i, 2]) - !(f[i, 2] % 2));}

Multiplicative with a(p^e) = d(e) if e is odd, and d(e)-1 if e is even, where d(k) is the number of divisors of k (A000005).
a(n) = 1 if and only if n is cubefree (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (d(k)+(k mod 2)-1)/p^k) = 1.1951330849... .

A340232 a(n) is the least number with exactly 2*n bi-unitary divisors.

Original entry on oeis.org

2, 6, 32, 24, 512, 96, 8192, 120, 131072, 1536, 2097152, 480, 33554432, 24576, 536870912, 840, 8589934592, 7776, 137438953472, 7680, 2199023255552, 6291456, 35184372088832, 3360, 562949953421312, 100663296, 9007199254740992, 122880, 144115188075855872, 124416

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

Every integer except 1 has an even number of bi-unitary divisors.

			a(1) = 2 since 2 is the least number with 2*1 = 2 bi-unitary divisors, 1 and 2.
a(2) = 6 since 6 is the least number with 2*2 = 4 bi-unitary divisors, 1, 2, 3 and 6.

Amiram Eldar, Table of n, a(n) for n = 1..62

Subsequence of A025487.
Cf. A222266, A286324, A293185.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340233 (exponential).

f[p_, e_] := If[OddQ[e], e + 1, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 10; s = Table[0, {max}]; c = 0; n = 2;  While[c < max, i = d[n]/2; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

A286324(a(n)) = 2*n and A286324(k) != 2*n for all k < a(n).

A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2d = 2k, where bsigma is the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

24, 30, 40, 42, 48, 54, 56, 66, 70, 78, 80, 88, 102, 104, 114, 120, 138, 150, 162, 174, 186, 222, 224, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 448, 474, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 672, 678, 720, 726, 762, 780, 786

Offset: 1

Amiram Eldar, May 18 2020

nonn

Equivalently, numbers that are equal to the sum of their proper bi-unitary divisors, with one of them taken with a minus sign.

Admirable numbers (A111592) that are exponentially odd (A268335) are also bi-unitary admirable numbers since all of their divisors are bi-unitary. Terms that are not exponentially odd are 48, 80, 150, 162, 294, 360, 420, 448, 540, 630, 660, 720, 726, 780, 832, 990, ...

			48 is in the sequence since 48 = 1 + 2 + 3 - 6 + 8 + 16 + 24 is the sum of its proper bi-unitary divisors with one of them, 6, taken with a minus sign.

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

The bi-unitary version of A111592.
Subsequence of A292982.
Cf. A188999, A222266, A268335, A328328, A334974.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); buDivQ[n_, 1] = True; buDivQ[n_, div_] := If[Mod[#2, #1] == 0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; buAdmQ[n_] := (ab = bsigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && buDivQ[n, ab/2]; Select[Range[1000], buAdmQ]

A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 88, 90, 96, 102, 104, 114, 120, 138, 150, 160, 162, 168, 174, 186, 192, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390, 402

Offset: 1

Amiram Eldar, May 27 2020

nonn

			6 is a term since its set of bi-unitary divisors, {1, 2, 3, 6}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 = 6.

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

The bi-unitary version of A083207.
Subsequence of A292982.
Cf. A188999, A222266, A290466, A323342, A335142, A335197.

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^3], bzQ]

cp's OEIS Frontend

A004709 Cubefree numbers: numbers that are not divisible by any cube > 1.

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A077609 Triangle in which n-th row lists infinitary divisors of n.

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A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.

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A286324 a(n) is the number of bi-unitary divisors of n.

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A322791 Irregular triangle read by rows in which the n-th row lists the exponential divisors (or e-divisors) of n.

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A286325 Bi-unitary harmonic numbers.

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A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2d = 2k, where bsigma is the sum of bi-unitary divisors function (A188999).