A286324 -id:A286324 - cp's OEIS frontend

A293185 Bi-unitary highly composite numbers: where the number of bi-unitary divisors of n (A286324) increases to a record.

1, 2, 6, 24, 96, 120, 480, 840, 3360, 7560, 30240, 83160, 272160, 332640, 1081080, 2993760, 4324320, 17297280, 38918880, 69189120, 73513440, 294053760, 661620960, 1176215040, 1396755360, 5587021440, 12570798240, 22348085760, 32125373280, 128501493120

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Analogous to highly composite numbers (A002182) with number of bi-unitary divisors (A286324) instead of number of divisors (A000005).
The first 12 terms are common with bi-unitary superabundant numbers (A292984).
The record numbers of bi-unitary divisors are 1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..122
Amiram Eldar, Table of n, a(n), A286324(a(n)) for n = 1..122

Cf. A002182, A286324, A292984.

f[p_, e_] := If[OddQ[e], (e + 1), e]; bdivnum[n_] := If[n == 1, 1, Times @@ (f @@@ FactorInteger[n])]; bm = 0; s = {}; Do[b1 = bdivnum [k]; If[b1 > bm, AppendTo[s, k]; bm = b1], {k, 1, 100000}]; s

a(18)-a(30) from Amiram Eldar, Dec 01 2018

A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 41, 45, 47, 51, 53, 57, 61, 65, 67, 75, 77, 81, 85, 89, 91, 99, 101, 107, 111, 115, 119, 123, 125, 129, 133, 141, 143, 151, 153, 157, 161, 165, 167, 175, 177, 181, 185, 189, 191, 199, 203, 211, 215, 219, 221

Offset: 1

Amiram Eldar, Jun 19 2018

nonn

The bi-unitary version of A006218 and A064608.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

Amiram Eldar, Table of n, a(n) for n = 1..10000
D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.

Cf. A001620, A006218, A064608, A286324, A306071, A306072.

fun[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; bdivnum[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Accumulate@ Array[bdivnum, {60}]

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) = sum(k=1, n, #biudivs(k)); \\ Michel Marcus, Jun 20 2018

a(n) = A*n*(log(n) + 2*gamma - 1 + B) + O(n^(1/2)*exp(-A * log(n)^(3/5) * log(log(n))^(-1/5))), where gamma = A001620, A = A306071 and B = A306072.

A344313 Number k such that k and k+1 have the same number of bi-unitary divisors (A286324).

Original entry on oeis.org

2, 3, 4, 14, 15, 20, 21, 26, 27, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 68, 74, 75, 76, 81, 85, 86, 91, 92, 93, 94, 98, 99, 104, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 145, 146, 147, 158, 171, 177, 187, 189, 201, 202, 205, 206, 212, 213, 214

Offset: 1

Amiram Eldar, May 14 2021

nonn

			2 is a term since A286324(2) = A286324(3) = 2.
14 is a term since A286324(14) = A286324(15) = 4.

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

Cf. A286324, A293183, A304463.

Similar sequences: A005237, A006049, A343819, A344312, A344314.

f[p_, e_] := If[OddQ[e], e + 1, e]; bd[1] = 1; bd[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], bd[#] == bd[# + 1] &]

A361059 Decimal expansion of the asymptotic mean of A000005(k)/A286324(k), the ratio between the number of divisors and the number of bi-unitary divisors.

Original entry on oeis.org

1, 1, 5, 8, 8, 5, 4, 5, 7, 2, 6, 5, 0, 3, 1, 2, 1, 0, 0, 1, 6, 4, 4, 8, 0, 1, 9, 6, 3, 9, 3, 1, 7, 5, 1, 4, 9, 0, 3, 9, 1, 0, 4, 3, 1, 8, 8, 5, 7, 3, 9, 5, 9, 6, 3, 4, 5, 2, 6, 1, 0, 6, 1, 5, 1, 4, 8, 2, 3, 3, 7, 9, 7, 4, 9, 3, 5, 4, 6, 4, 9, 0, 6, 6, 6, 5, 1, 3, 9, 2, 1, 7, 9, 2, 9, 5, 4, 7, 3, 9, 6, 2, 5, 7, 3

Offset: 1

Amiram Eldar, Mar 01 2023

			1.158854572650312100164480196393175149039104318857395...

Eric Weisstein's World of Mathematics, Biunitary Divisor.

Cf. A000005, A286324, A361060 (mean of the inverse ratio).

Cf. A307869 (unitary analog), A308043.

$MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)*Log[1 - 1/p^2]/(2*p); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n], {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 106][[1]]

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A286324(k).

Equals Product_{p prime} (1 - (p-1)*log(1 - 1/p^2)/(2*p)).

A361060 Decimal expansion of the asymptotic mean of A286324(k)/A000005(k), the ratio between the number of bi-unitary divisors and the number of divisors.

Original entry on oeis.org

9, 0, 1, 2, 4, 1, 8, 0, 6, 8, 2, 6, 4, 8, 2, 2, 5, 5, 1, 3, 9, 1, 9, 7, 4, 8, 5, 0, 9, 4, 3, 8, 7, 5, 5, 8, 9, 8, 2, 8, 1, 1, 5, 3, 3, 8, 2, 1, 7, 8, 7, 6, 2, 8, 7, 6, 2, 6, 1, 6, 1, 2, 0, 6, 3, 0, 9, 0, 7, 3, 4, 3, 7, 3, 3, 1, 8, 6, 0, 8, 3, 7, 9, 3, 6, 3, 5, 5, 9, 5, 4, 0, 8, 6, 0, 1, 0, 5, 2, 4, 5, 6, 4, 9, 8

Offset: 0

Amiram Eldar, Mar 01 2023

			0.901241806826482255139197485094387558982811533821787...

Eric Weisstein's World of Mathematics, Biunitary Divisor.

Cf. A000005, A286324, A361059 (mean of the inverse ratio).

Cf. A307869, A308043 (unitary analog).

$MaxExtraPrecision = 1000; m = 1000; f[p_] := 2 - 1/p - (p - 1)*Log[(p + 1)/(p - 1)]/2; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n], {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A286324(k)/A000005(k).

Equals Product_{p prime} (2 - 1/p - (p-1)*log((p+1)/(p-1))/2).

A293618 Numbers n that equal the sum of their first k consecutive aliquot bi-unitary divisors, but not all of them (i.e k < A286324(n)-1).

Original entry on oeis.org

24, 360, 432, 1344, 2016, 19440, 45360, 68544, 714240, 864000, 1468800, 1571328, 1900800, 2391120, 2888704, 3057600, 4586400, 5241600, 103194000

Offset: 1

Amiram Eldar, Oct 13 2017

The bi-unitary version of Erdős-Nicolas numbers (A194472).

If all the aliquot bi-unitary divisors are permitted (i.e. k <= A286324(n)-1), then the 3 bi-unitary perfect numbers, 6, 60 and 90, are included.

			24 is in the sequence since its aliquot bi-unitary divisors are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24.

Cf. A188999, A194472, A222266, A286324.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[bdiv[n], -2]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

A268335 Exponentially odd numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Vladimir Shevelev, Feb 01 2016

nonn

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463.
Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016
Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018
The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, Distribution of unitarily k-free integers, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
A268335_list = list(islice(A268335_gen(),20)) # Chai Wah Wu, Jun 22 2023

Sum_{a(n)<=x} 1 = C*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 C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A222266 Irregular triangle which lists the bi-unitary divisors of n in row 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, 2, 8, 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, 2, 3, 5, 6, 10, 15, 30, 1, 31, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34, 1, 5, 7, 35

Offset: 1

R. J. Mathar, May 05 2013

The bi-unitary divisors of n are the divisors of n such that the largest common unitary divisor of d and n/d is 1, indicated by A165430.
The first difference from the triangle A077609 is in row n=16.
The concept of bi-unitary divisors was introduced by Suryanarayana (1972). - Amiram Eldar, Mar 09 2024

			The table starts
  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, 2, 8, 16;
  1, 17;

Michael De Vlieger, Table of n, a(n) for n = 1..13171 (rows 1 <= n <= 2000).
D. Suryanarayana, The number of bi-unitary divisors of an integer, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics, Vol 251, Springer, Berlin, Heidelberg, 1972.

Cf. A077609, A165430, A188999 (row sums), A286324 (row lengths).

# Return set of unitary divisors of n.
A077610_row := proc(n)
    local u,d ;
    u := {} ;
    for d in numtheory[divisors](n) do
        if igcd(n/d,d) = 1 then
            u := u union {d} ;
        end if;
    end do:
    u ;
end proc:
# true if d is a bi-unitary divisor of n.
isbiudiv := proc(n,d)
    if n mod d = 0 then
        A077610_row(d) intersect A077610_row(n/d) ;
        if % = {1} then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
# Return set of bi-unitary divisors of n
biudivs := proc(n)
    local u,d ;
    u := {} ;
    for d in numtheory[divisors](n) do
        if isbiudiv(n,d) then
            u := u union {d} ;
        end if;
    end do:
    u ;
end proc:
for n from 1 to 35 do
    print(op(biudivs(n))) ;
end do:

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[Function[d, Union@ Flatten@ Select[Transpose@ {d, n/d}, Last@ Intersection[f@ #1, f@ #2] == 1 & @@ # &]]@ Select[Divisors@ n, # <= Floor@ Sqrt@ n &], {n, 35}] (* Michael De Vlieger, May 07 2017 *)

isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
row(n) = {my(d = divisors(n), f = factor(n), bdiv = []); for(i=1, #d, if(isbdiv(f, d[i]), bdiv = concat(bdiv, d[i]))); bdiv; } \\ Amiram Eldar, Mar 24 2023

A322483 The number of semi-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Dec 11 2018

The notion of semi-unitary divisor was introduced by Chidambaraswamy in 1967.
A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1. In terms of the relation defined in A322482, d is the largest divisor of n such that T(d, n/d) = 1 (the largest divisor d that is semiprime to n/d).
The number of divisors of n that are exponentially odd numbers (A268335). - Amiram Eldar, Sep 08 2023

			The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their number is 3, thus a(8) = 3.

J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, Journal of the Australian Mathematical Society, Vol. 23, No. 1 (1977), pp. 9-27.
Pentti Haukkanen, Basic properties of the bi-unitary convolution and the semi-unitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305-315.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions. Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of unitarily k-free divisors of an integer, Journal of the Australian Mathematical Society, Vol. 21, No. 1 (1976), pp. 19-35.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.

Cf. A000005, A005117, A019554, A034444, A268335, A286324, A322482.

Cf. A001620, A056671, A343443, A365488, A365491.

f[p_, e_] := Floor[(e+3)/2]; sud[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[sud, 100]

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = (f[k,2]+3)\2; f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Dec 14 2018

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1-X^2) * (1 + X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 06 2023

Multiplicative with a(p^e) = floor((e+3)/2).
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
a(n) = Sum_{d|n} mu(d/gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
a(n) = A000005(A019554(n)) (the number of divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 02 2023
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Let f(s) = Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444...,
f'(1) = f(1) * Sum_{p prime} (4*p-3) * log(p) / (p^3 - 2*p + 1) = 0.808661108949590913395... and gamma is the Euler-Mascheroni constant A001620. (End)

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;

cp's OEIS Frontend

A293185 Bi-unitary highly composite numbers: where the number of bi-unitary divisors of n (A286324) increases to a record.

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A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

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A344313 Number k such that k and k+1 have the same number of bi-unitary divisors (A286324).

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A361059 Decimal expansion of the asymptotic mean of A000005(k)/A286324(k), the ratio between the number of divisors and the number of bi-unitary divisors.

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A361060 Decimal expansion of the asymptotic mean of A286324(k)/A000005(k), the ratio between the number of bi-unitary divisors and the number of divisors.

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A293618 Numbers n that equal the sum of their first k consecutive aliquot bi-unitary divisors, but not all of them (i.e k < A286324(n)-1).

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A268335 Exponentially odd numbers.

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A222266 Irregular triangle which lists the bi-unitary divisors of n in row n.

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