A018892 -id:A018892 - cp's OEIS frontend

A252505 Number of biquadratefree (4th power free) divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 8, 2, 6, 6, 9

Offset: 1

Geoffrey Critzer, Mar 21 2015

Equivalently, a(n) is the number of divisors of n that are in A046100.
a(n) is also the number of divisors d such that the greatest common square divisor of d and n/d is 1.
The number of divisors d of n such that gcd(d, n/d) is squarefree. - Amiram Eldar, Aug 25 2023

			a(16) = 4 because there are 4 divisors of 16 that are 4th power free: 1,2,4,8.
a(16) = 4 because there are 4 divisors d of 16 such that the greatest common square divisor of d and 16/d is 1: 1,2,8,16.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A252505, Sequence Machine.
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046100 (biquadratefree numbers).
Cf. A034444 (squarefree divisors), A073184 (cubefree divisors).
Cf. A001620.
Cf. A000005, A058035, A307430.
Also obtained as a Dirichlet convolution of the following pairs: A034444 and A227291, A007427 and A286779, A008966 and A323308, A048691 and A363552, A271102 and A322327, A307445 and A370296, and A018892 and A378214 (conjectured).

Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]

isA046100(n) = (n==1) || vecmax(factor(n)[, 2])<4;
a(n) = {d = divisors(n); sum(i=1, #d, isA046100(d[i]));} \\ Michel Marcus, Mar 22 2015

a(n) = vecprod(apply(x->min(x, 3) + 1, factor(n)[, 2])); \\ Amiram Eldar, Aug 25 2023

Dirichlet g.f.: zeta(s)^2/zeta(4*s).
Sum_{k=1..n} a(k) ~ 90*n/Pi^4 * (log(n) - 1 + 2*gamma - 360*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} mu(gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020
From Antti Karttunen, May 14 2025: (Start)
Following formulas have been generated for this sequence by Sequence Machine:
a(n) = A000005(A058035(n)).
a(n) = Sum_{d|n} A307430(d).
a(n) = Sum_{d|n} A034444(d)*A227291(n/d).
a(n) = Sum_{d|n} A007427(d)*A286779(n/d).
a(n) = Sum_{d|n} A008966(d)*A323308(n/d).
a(n) = Sum_{d|n} A048691(d)*A363552(n/d).
a(n) = Sum_{d|n} A271102(d)*A322327(n/d).
a(n) = Sum_{d|n} A307445(d)*A370296(n/d).
a(n) = Sum_{d|n} A018892(d)*A378214(n/d). [Conjectured]
(End)

A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 4, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 7, 4, 2, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 3, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 4, 4, 1, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 4, 3, 2, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

Jud McCranie, Nov 07 2021

The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with row 2, column 1) and the columns correspond to the numerators:
                    x=1  2  3  4  5       rationals x/y:
  Row 1 (y=2):  1                   1/2
  Row 2 (y=3):  1, 1                1/3, 2/3
  Row 3 (y=4):  2, 1, 1             1/4, 2/4, 3/4
  Row 4 (y=5):  1, 1, 1, 0          1/5, 2/5, 3/5, 4/5
  Row 5 (y=6):  4, 1, 1, 1, 1       1/6, 2/6, 3/6, 4/6, 5/6
Alternatively, order the rational numbers, x/y, 0 < x/y < 1, in this order: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ... For example, in this ordering, the sixth rational number is 3/4. The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).
A018892 is a subsequence (for x/y = 1/n).

			The fourth rational number is 1/4, 1/4 = 1/5 + 1/20 = 1/6 + 1/12, so a(4)=2.

Jud McCranie, Table of n, a(n) for n = 1..990

Cf. A002260, A003057.

Columns: A018892 (x=1), A046079 (x=2).

A063520 Sum divides product: number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t).

Original entry on oeis.org

1, 3, 6, 5, 8, 8, 8, 14, 13, 9, 14, 17, 8, 18, 23, 18, 14, 17, 13, 33, 23, 10, 19, 36, 15, 22, 32, 22, 19, 26, 17, 39, 24, 18, 50, 45, 8, 22, 39, 38, 22, 27, 13, 50, 45, 16, 27, 52, 24, 39, 38, 27, 20, 50, 45, 72, 24, 12, 31, 58, 15, 28, 69, 45, 49, 39, 12, 52, 40, 33, 33, 66, 12, 33, 64

Offset: 1

Jud McCranie and Vladeta Jovovic, Aug 01 2001

nonn

Number of solutions (r,s) in positive integers to the equation rs = n(r+s) is tau(n^2), cf. A048691. Number of solutions (r,s), r>=s>0, to the equation rs = n(r+s) is (tau(n^2)+1)/2, cf. A018892.

Conjecturally, includes all positive integers except 2, 4, 7 and 11 - David W. Wilson

			There are 8 such solutions to rst = 5(r+s+t): (5, 4, 3), (7, 5, 2), (10, 4, 2), (11, 10, 1), (15, 8, 1), (20, 7, 1), (25, 3, 2), (35, 6, 1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. J. Pelling, Problem 10745, Amer. Math. Monthly, vol. 106 (1999), p. 587.
M. J. Pelling and F. W. Roush, The Sum Divides the Product: Problem 10745, Amer. Math. Monthly, vol. 108, (no. 7, Aug. 2001), pp. 668-669. [Gives upper bound]

Cf. A018892, A004194, A063525.

(* Assuming s <= 2n and t <= n*(n+2) *) redu[n_] := Reap[ Do[ red = Reduce[0 < r <= s <= t && r*s*t == n*(r+s+t), r, Integers]; If[red =!= False, Sow[{r, s, t} /. ToRules[red] ] ], {s, 1, 2*n}, {t, s, n*(n+2)}] ][[2, 1]]; a[n_] := redu[n] // Length; a[1] = 1; Table[ Print[n, " ", an = a[n]]; an, {n, 1, 75}] (* Jean-François Alcover, Feb 22 2013 *)

a(n)=sum(t=1,sqrtint(3*n),sum(s=t,sqrtint(n^2+t)+n,my(N=n*(s+t), D=s*t-n);D&&denominator(N/D)==1&&N/D>=s)) \\ Charles R Greathouse IV, Feb 22 2013

More terms from David W. Wilson, Aug 01 2001

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A015995 at a(210) = 88, A015995(210) = 86.

			For example, the a(n) subsets for n = 1, 2, 4, 6, 8, 12, 16, 24 are:
  {1}  {1}    {1}    {1}      {1}    {1}      {1}     {1}
       {1,2}  {1,2}  {1,2}    {1,2}  {1,2}    {1,2}   {1,2}
              {1,4}  {1,3}    {1,4}  {1,3}    {1,4}   {1,3}
                     {1,6}    {1,8}  {1,4}    {1,8}   {1,4}
                     {2,3}           {1,6}    {1,16}  {1,6}
                     {1,2,3}         {2,3}            {1,8}
                                     {3,4}            {2,3}
                                     {1,12}           {3,4}
                                     {1,2,3}          {3,8}
                                     {1,3,4}          {1,12}
                                                      {1,24}
                                                      {1,2,3}
                                                      {1,3,4}
                                                      {1,3,8}

The case of pairs is A063647.
The case of triples is A066620.
The version with empty sets and singletons is A225520.
A version for prime indices is A304711.
The version for strict integer partitions is A305713.
The version for subsets of {1..n} is A320426 = A276187 + 1.
The version for binary indices is A326675.
The version for integer partitions is A327516.
The version for standard compositions is A333227.
The maximal case is A343652.
The case without 1's is A343653.
The case without 1's with singletons is A343654.
The maximal case without 1's is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A062319, A067824, A076078, A084422, A187106, A282935, A285572, A304709, A320423, A337485, A343659.

Table[Length[Select[Subsets[Divisors[n]],CoprimeQ@@#&]],{n,100}]

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

A343654 Number of pairwise coprime sets of divisors > 1 of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A100565 at a(210) = 52, A100565(210) = 51.

			The a(n) sets for n = 1, 2, 4, 6, 8, 12, 24, 30, 32, 36, 48:
  {}  {}   {}   {}     {}   {}     {}     {}       {}    {}     {}
      {2}  {2}  {2}    {2}  {2}    {2}    {2}      {2}   {2}    {2}
           {4}  {3}    {4}  {3}    {3}    {3}      {4}   {3}    {3}
                {6}    {8}  {4}    {4}    {5}      {8}   {4}    {4}
                {2,3}       {6}    {6}    {6}      {16}  {6}    {6}
                            {12}   {8}    {10}     {32}  {9}    {8}
                            {2,3}  {12}   {15}           {12}   {12}
                            {3,4}  {24}   {30}           {18}   {16}
                                   {2,3}  {2,3}          {36}   {24}
                                   {3,4}  {2,5}          {2,3}  {48}
                                   {3,8}  {3,5}          {2,9}  {2,3}
                                          {5,6}          {3,4}  {3,4}
                                          {2,15}         {4,9}  {3,8}
                                          {3,10}                {3,16}
                                          {2,3,5}

The version for partitions is A007359.
The version for subsets of {1..n} is A084422.
The case of pairs is A089233.
The version with 1's is A225520.
The maximal case is A343652.
The case without empty sets or singletons is A343653.
The maximal case without singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A187106, A276187, and A320426 count other types of pairwise coprime sets.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A051026, A062319, A074206, A087087, A101268, A285572, A305713, A320423, A326675, A337485, A343655.

pwcop[y_]:=And@@(GCD@@#1==1&)/@Subsets[y,{2}];
Table[Length[Select[Subsets[Rest[Divisors[n]]],pwcop]],{n,100}]

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 7, 9, 9, 10, 10, 12, 16, 19, 19, 20, 20, 22, 28, 32, 32, 33, 54, 61, 77, 84, 84, 85, 85, 94, 112, 123, 158, 161, 161, 176, 206, 212, 212, 214, 214, 229, 241, 260, 260, 263, 417, 428, 490, 521, 521, 526, 655, 674, 764, 818, 818, 820, 820, 874, 918, 975, 1182, 1189, 1189

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

For this sequence, it does not matter whether singletons are considered pairwise coprime.
For n > 2, also the number of maximal pairwise coprime subsets of {2..n}.
For each prime p <= n, p divides exactly one element of each maximal subset. - Bert Dobbelaere, May 04 2021

			The a(1) = 1 through a(9) = 7 subsets:
  {1}  {12}  {123}  {123}  {1235}  {156}   {1567}   {1567}   {1567}
                    {134}  {1345}  {1235}  {12357}  {12357}  {12357}
                                   {1345}  {13457}  {13457}  {12579}
                                                    {13578}  {13457}
                                                             {13578}
                                                             {14579}
                                                             {15789}

Bert Dobbelaere, Table of n, a(n) for n = 1..500
Bert Dobbelaere, Python program

The case of pairs is A015614.
The case of triples is A015617.
The non-maximal version counting empty sets and singletons is A084422.
The non-maximal version counting singletons is A187106.
The non-maximal version is A320426(n) = A276187(n) + 1.
The version for indivisibility instead of coprimality is A326077.
The version for sets of divisors is A343652.
The version for sets of divisors > 1 is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A007360, A067824, A087087, A225520, A324837, A325683, A325859, A326358, A326496, A326675, A333227, A343653, A343655.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],CoprimeQ@@#&]]],{n,15}]

More terms from Bert Dobbelaere, May 04 2021

A015702 Numbers k where phi(k) + sigma(k) increases to a record value.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 40, 42, 48, 56, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 280, 288, 300, 324, 336, 360, 420, 480, 504, 540, 576, 600, 648, 660, 672, 720, 840, 960, 1008, 1080, 1200, 1260, 1440

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1288 (terms < 10^12)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A000203, A000010, A018892, A018894, A065387.

seq = {}; sm = 0; s = 0; Do[s = EulerPhi[n] + DivisorSigma[1, n];
If[s > sm, sm = s; AppendTo[seq, n]], {n, 1, 1500}]; seq (* Amiram Eldar, Dec 05 2018 *)
DeleteDuplicates[Table[{n,EulerPhi[n]+DivisorSigma[1,n]},{n,1500}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 13 2023 *)

f(n)=eulerphi(n=factor(n))+sigma(n)
r=0;for(n=1,1e6,t=f(n); if(t>r,r=t; print1(n", "))) \\ Charles R Greathouse IV, Nov 27 2013

A015995 a(n) = (tau(n^3)+2)/3.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2, 18, 5, 6, 2, 38, 6, 6

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A004194, A018892, A015996, A015999, A016001, A016002, A016003, A016005-A016009, A016012, A016017, A016018, A016020, A016025, A048785.

Table[(DivisorSigma[0,n^3]+2)/3,{n,90}] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=(numdiv(n^3)+2)/3 \\ Charles R Greathouse IV, May 01 2013

a(n)=my(f=factor(n)[,2]);prod(i=1,#f,3*f[i]+1)\3+1 \\ Charles R Greathouse IV, May 01 2013

a(n) = (2+A048785(n))/3. - R. J. Mathar, May 07 2021

Definition corrected by Vladeta Jovovic, Sep 03 2005

A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

First differs from A066620 at a(210) = 36, A066620(210) = 35.

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {2,3}    {2,3}  {2,3}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,5}    {2,9}  {2,5}    {2,9}  {3,4}
                {3,8}  {3,5}    {3,4}  {3,4}    {3,4}  {3,8}
                       {5,6}    {4,9}  {3,5}    {3,8}  {3,16}
                       {2,15}          {4,5}    {4,9}  {3,32}
                       {3,10}          {5,6}    {8,9}
                       {2,3,5}         {2,15}
                                       {3,10}
                                       {3,20}
                                       {4,15}
                                       {5,12}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The version with 1's, empty sets, and singletons is A225520.
The version for subsets of {1..n} is A320426.
The version for strict partitions is A337485.
The version for compositions is A337697.
The version for prime indices is A337984.
The maximal case with 1's is A343652.
The version with empty sets is a(n) + 1.
The version with singletons is A343654(n) - 1.
The version with empty sets and singletons is A343654.
The version with 1's is A343655.
The maximal case is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A343659 counts maximal pairwise coprime subsets of {1..n}.
Cf. A007359, A067824, A074206, A076078, A084422, A187106, A285572, A324837, A326675, A327516, A338315.

Table[Length[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]],{n,100}]

cp's OEIS Frontend

A252505 Number of biquadratefree (4th power free) divisors of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A063520 Sum divides product: number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A343654 Number of pairwise coprime sets of divisors > 1 of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A015702 Numbers k where phi(k) + sigma(k) increases to a record value.