A018892 -id:A018892 - cp's OEIS frontend

A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

1, 3, 3, 6, 3, 10, 3, 10, 6, 10, 3, 21, 3, 10, 10, 15, 3, 21, 3, 21, 10, 10, 3, 36, 6, 10, 10, 21, 3, 36, 3, 21, 10, 10, 10, 45, 3, 10, 10, 36, 3, 36, 3, 21, 21, 10, 3, 55, 6, 21, 10, 21, 3, 36, 10, 36, 10, 10, 3, 78, 3, 10, 21, 28, 10, 36, 3, 21, 10, 36, 3, 78

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Length of row n in triangle A187207. - Omar E. Pol, Aug 07 2011

Number of pairs of even divisors of 2n, (d1,d2), such that d1<=d2. - Wesley Ivan Hurt, Aug 24 2020

			For n = 4; tau(4) = 3; a(4) = 1+2+3 = 6.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A000005 (tau), A000217 (triangular numbers).
Cf. A184387, A184388, A184390, A184391, A130674.
Cf. A035116, A066446.
Cf. A187207, A135539, A337362, A129308.

a184389 = a000217 . a000005'  -- Reinhard Zumkeller, Sep 08 2015

A184389 := proc(n)
    A000217(numtheory[tau](n)) ;
end proc: # R. J. Mathar, Oct 04 2014

((#+1)#)/2&/@DivisorSigma[0,Range[80]] (* Harvey P. Dale, Feb 27 2013 *)

a(n) = my(nd=numdiv(n)); nd*(nd+1)/2; \\ Michel Marcus, Jun 25 2016

a(n) = A000217(A000005(n)) = (1/2)*A000005(n)*(A000005(n)+1).
a(n) = A066446(n) + A000005(n) = A035116(n) - A066446(n). - Reinhard Zumkeller, Sep 08 2015
Dirichlet g.f.: zeta(s)^2*(zeta(s)^2 + zeta(2*s))/(2*zeta(2*s)). - Ilya Gutkovskiy, Jun 25 2016
a(n) = Sum_{d1|(2*n), d2|(2*n), d1 and d2 even, d1<=d2} 1. - Wesley Ivan Hurt, Aug 24 2020
a(n) = Sum_{d|n} A018892(d). - Daniel Suteu, Jan 08 2021
a(n) = Sum_{d|n} A135539(n,d). - Ridouane Oudra, May 29 2025
a(n) = A337362(n) + A129308(n). - Ridouane Oudra, May 30 2025

A007875 Number of ways of writing n as p*q, with p <= q, gcd(p, q) = 1.

Original entry on oeis.org

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

Offset: 1

Victor Ufnarovski

a(n), n >= 2, is the number of divisor products in the numerator as well as denominator of the unique representation of n in terms of divisor products. See the W. Lang link under A007955, where a(n)=l(n) in Table 1. - Wolfdieter Lang, Feb 08 2011
Record values are the binary powers, occurring at primorial positions except at 2: a(A002110(0))=A000079(0), a(A002110(n+1))=A000079(n) for n > 0. - Reinhard Zumkeller, Aug 24 2011
For n > 1: a(n) = (A000005(n) - A048105(n)) / 2; number of ones in row n of triangle in A225817. - Reinhard Zumkeller, Jul 30 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Larry Bates and Peter Gibson, A geometry where everything is better than nice, arXiv:1603.06622 [math.DG], (21-March-2016); see page 2.
Vaclav Kotesovec, Graph - the asymptotic ratio.

Cf. A000005, A000010, A000079, A001221, A001620, A002110, A007955, A034444, A048105, A074962.

a007875 = length . filter (> 0) . a225817_row
-- Reinhard Zumkeller, Jul 30 2013, Aug 24 2011

A007875 := proc(n)
    if n = 1 then
        1;
    else
        2^(A001221(n)-1) ;
    end if;
end proc: # R. J. Mathar, May 28 2016

a[n_] := With[{r = Reduce[1 <= p <= q <= n && n == p*q && GCD[p, q] == 1, {p, q}, Integers]}, If[Head[r] === And, 1, Length[r]]]; Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Nov 02 2011 *)
a[n_] := EulerPhi[2^PrimeNu[n]]; Array[a, 105] (* Robert G. Wilson v, Apr 10 2012 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[MoebiusMu[k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)

a(n)=ceil((1<Charles R Greathouse IV, Nov 02 2011

a(n) = (1/2)*Sum_{ d divides n } abs(mu(d)) = 2^(A001221(n)-1) = A034444(n)/2, n > 1. - Vladeta Jovovic, Jan 25 2002
a(n) = phi(2^omega(n)) = A000010(2^A001221(n)). - Enrique Pérez Herrero, Apr 10 2012
Sum_{k=1..n} a(k) ~ 3*n*((log(n) + (2*gamma - 1))/ Pi^2 - 12*(zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. Equivalently, Sum_{k=1..n} a(k) ~ 3*n*(log(n) + 24*log(A) - 1 - 2*log(2*Pi)) / Pi^2, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 30 2019
a(n) = Sum_{d|n} mu(d) * A018892(n/d). - Daniel Suteu, Jan 08 2021
Dirichlet g.f.: (zeta(s)^2/zeta(2*s) + 1)/2. - Amiram Eldar, Sep 09 2023

A015999 a(n) = (tau(n^5) + 4)/5.

Original entry on oeis.org

1, 2, 2, 3, 2, 8, 2, 4, 3, 8, 2, 14, 2, 8, 8, 5, 2, 14, 2, 14, 8, 8, 2, 20, 3, 8, 4, 14, 2, 44, 2, 6, 8, 8, 8, 25, 2, 8, 8, 20, 2, 44, 2, 14, 14, 8, 2, 26, 3, 14, 8, 14, 2, 20, 8, 20, 8, 8, 2, 80, 2, 8, 14, 7, 8, 44, 2, 14, 8, 44, 2, 36, 2, 8, 14, 14, 8, 44, 2, 26, 5, 8, 2, 80, 8, 8

Offset: 1

Robert G. Wilson v

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

Cf. A000005, A018892, A015995, A015996, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016012.

with(numtheory): A015999:=n->(tau(n^5)+4)/5: seq(A015999(n), n=1..80); # Wesley Ivan Hurt, Apr 10 2015

(DivisorSigma[0, Range[80]^5]+4)/5 (* Wesley Ivan Hurt, Apr 10 2015 *)

A015999(n) = (numdiv(n^5)+4)/5;
for(n=1, 10000, write("b015999.txt", n, " ", A015999(n)));
\\ Antti Karttunen, Jan 17 2017

from sympy import divisor_count
def a(n): return (divisor_count(n**5) + 4)//5
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 14 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A089233 Number of coprime pairs 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, 6, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 6, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 11, 0, 1, 2, 0, 1, 6, 0, 2, 1, 6, 0, 6, 0, 1, 2, 2, 1, 6, 0, 4, 0, 1, 0, 11, 1, 1, 1, 3, 0, 11, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 6, 0, 3, 6

Offset: 1

Reinhard Zumkeller, Dec 11 2003

nonn

Also the number of divisors of n^2 which do not divide n and which are less than n. See link for proof. - Andrew Weimholt, Dec 06 2009
a(A000961(n)) = 0; a(A006881(n)) = 1; a(A054753(n)) = 2; a(A065036(n)) = 3. - Robert G. Wilson v, Dec 16 2009
First occurrence of k beginning with 0: 1, 6, 12, 24, 36, 96, 30, 384, 144, 216, 288, 60, 432, 24576, 1152, 864, 120, 393216, 1728, 1572864, 180, 240, 18432, 25165824, 5184, 210, 480, 13824, 10368, 360, 15552, 960, 20736, 55296, 1179648, 31104, 900, ..., . Except for 1, each is divisible by 6. Also the first occurrence of k must occur at or before 6*2^(n-1). - Robert G. Wilson v, Dec 16 2009
a(3*2^n) = n; if x = 2^n, then a(x) = a(2*x); and if x is not a power of two, then a(x) = y then a(2*x) > y. - Robert G. Wilson v, Dec 16 2009
a(n) = 0 iff n is a prime power. - Franklin T. Adams-Watters, Aug 20 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew Weimholt, Proof of an alternative characterization

a089233 n = sum $ [a063524 $ gcd u v | let ds = tail $ a027750_row n,
                                       u <- ds, v <- dropWhile (<= u) ds]
-- Reinhard Zumkeller, Sep 04 2013

[(NumberOfDivisors(n^2)-1)/2 - NumberOfDivisors(n)+1: n in [1..100]]; // Vincenzo Librandi, Dec 23 2018

a[n_] := (DivisorSigma[0, n^2] - 1)/2 - DivisorSigma[0, n] + 1; Array[a, 104] (* Robert G. Wilson v, Dec 16 2009 *)

a(n) = (numdiv(n^2)-1)/2 - numdiv(n) + 1; \\ Michel Marcus, Feb 17 2016

a(n) = #{(x,y): 1 < x < y, x|n, y|n and gcd(x, y) = 1}.
a(n) = A063647(n) - A000005(n) + 1.
a(n) = A018892(n) - A000005(n). - Franklin T. Adams-Watters, Aug 20 2013

A320426 Number of nonempty pairwise coprime subsets of {1,...,n}, where a single number is not considered to be pairwise coprime unless it is equal to 1.

Original entry on oeis.org

1, 2, 5, 8, 19, 22, 49, 64, 95, 106, 221, 236, 483, 530, 601, 712, 1439, 1502, 3021, 3212, 3595, 3850, 7721, 7976, 11143, 11878, 14629, 15460, 30947, 31202, 62433, 69856, 76127, 80222, 89821, 91612, 183259, 192602, 208601, 214232, 428503, 431574, 863189

Offset: 1

Gus Wiseman, Jan 08 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1.

			The a(4) = 8 subsets of {1,2,3,4} are {1}, {1,2}, {1,3}, {1,4}, {2,3}, {3,4}, {1,2,3}, {1,3,4}. - _Michael B. Porter_, Jan 12 2019
From _Gus Wiseman_, May 09 2021: (Start)
The a(2) = 2 through a(6) = 22 sets:
   {1}     {1}      {1}       {1}        {1}
  {1,2}   {1,2}    {1,2}     {1,2}      {1,2}
          {1,3}    {1,3}     {1,3}      {1,3}
          {2,3}    {1,4}     {1,4}      {1,4}
         {1,2,3}   {2,3}     {1,5}      {1,5}
                   {3,4}     {2,3}      {1,6}
                  {1,2,3}    {2,5}      {2,3}
                  {1,3,4}    {3,4}      {2,5}
                             {3,5}      {3,4}
                             {4,5}      {3,5}
                            {1,2,3}     {4,5}
                            {1,2,5}     {5,6}
                            {1,3,4}    {1,2,3}
                            {1,3,5}    {1,2,5}
                            {1,4,5}    {1,3,4}
                            {2,3,5}    {1,3,5}
                            {3,4,5}    {1,4,5}
                           {1,2,3,5}   {1,5,6}
                           {1,3,4,5}   {2,3,5}
                                       {3,4,5}
                                      {1,2,3,5}
                                      {1,3,4,5}
(End)

The case of pairs is A015614.
The case with singletons is A187106.
The version without singletons (except {1}) is A276187.
Row sums of A320436.
The version for divisors > 1 is A343654.
The version for divisors without singletons is A343655.
The maximal version is A343659.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1...n}.
A087087 ranks pairwise coprime subsets of {1...n}.
A326675 ranks pairwise coprime non-singleton subsets of {1...n}.
Cf. A007359, A007360, A066620, A089233, A100565, A225520, A325683, A326359, A337485, A343652.

Table[Length[Select[Subsets[Range[n]],CoprimeQ@@#&]],{n,10}]

a(n) = A187106(n) - n + 1 = A084422(n) - n.

a(n) = A276187(n) + 1. - Gus Wiseman, May 08 2021

a(25)-a(43) from Alois P. Heinz, Jan 08 2019

A015996 (tau(n^4) + 3)/4, where tau = A000005.

Original entry on oeis.org

1, 2, 2, 3, 2, 7, 2, 4, 3, 7, 2, 12, 2, 7, 7, 5, 2, 12, 2, 12, 7, 7, 2, 17, 3, 7, 4, 12, 2, 32, 2, 6, 7, 7, 7, 21, 2, 7, 7, 17, 2, 32, 2, 12, 12, 7, 2, 22, 3, 12, 7, 12, 2, 17, 7, 17, 7, 7, 2, 57, 2, 7, 12, 7, 7, 32, 2, 12, 7, 32, 2, 30, 2, 7, 12, 12, 7, 32, 2, 22, 5, 7, 2, 57, 7, 7

Offset: 1

Robert G. Wilson v

nonn

If n is prime, a(n) = 2 since a(p) = (tau(p^4)+3)/4 = (5+3)/4 = 2. - Wesley Ivan Hurt, Nov 16 2013

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

Cf. A000005, A018892, A015995, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A016012, A016020.

A015996 := proc(n)
        (numtheory[tau](n^4)+3)/4 ;
end proc; # R. J. Mathar, May 09 2013

Table[(DivisorSigma[0, n^4] + 3)/4, {n, 100}] (* Wesley Ivan Hurt, Nov 16 2013 *)

A015996(n) = (numdiv(n^4)+3)/4;
for(n=1, 10000, write("b015996.txt", n, " ", A015996(n)));
\\ Antti Karttunen, Jan 17 2017

a(n) = (A000005(n^4) + 3)/4.

Definition corrected by Vladeta Jovovic, Sep 03 2005

A343652 Number of maximal pairwise coprime sets of divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:
  {6}    {30}     {6}    {30}     {30}
  {12}   {2,15}   {12}   {60}     {60}
  {2,3}  {3,10}   {18}   {2,15}   {120}
  {3,4}  {5,6}    {36}   {3,10}   {2,15}
         {2,3,5}  {2,3}  {3,20}   {3,10}
                  {2,9}  {4,15}   {3,20}
                  {3,4}  {5,6}    {3,40}
                  {4,9}  {5,12}   {4,15}
                         {2,3,5}  {5,6}
                         {3,4,5}  {5,12}
                                  {5,24}
                                  {8,15}
                                  {2,3,5}
                                  {3,4,5}
                                  {3,5,8}

			The a(n) sets for n = 12, 30, 36, 60, 120:
  {1,6}    {1,30}     {1,6}    {1,30}     {1,30}
  {1,12}   {1,2,15}   {1,12}   {1,60}     {1,60}
  {1,2,3}  {1,3,10}   {1,18}   {1,2,15}   {1,120}
  {1,3,4}  {1,5,6}    {1,36}   {1,3,10}   {1,2,15}
           {1,2,3,5}  {1,2,3}  {1,3,20}   {1,3,10}
                      {1,2,9}  {1,4,15}   {1,3,20}
                      {1,3,4}  {1,5,6}    {1,3,40}
                      {1,4,9}  {1,5,12}   {1,4,15}
                               {1,2,3,5}  {1,5,6}
                               {1,3,4,5}  {1,5,12}
                                          {1,5,24}
                                          {1,8,15}
                                          {1,2,3,5}
                                          {1,3,4,5}
                                          {1,3,5,8}

The case of pairs is A063647.
The case of triples is A066620.
The non-maximal version counting empty sets and singletons is A225520.
The non-maximal version with no 1's is A343653.
The non-maximal version is A343655.
The version for subsets of {1..n} is A343659.
The case without 1's or singletons 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.
A084422, A187106, A276187, and A320426 count pairwise coprime sets.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A324837 counts minimal subsets of {1...n} with least common multiple n.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A005361, A007359, A051026, A062319, A067824, A074206, A146291, A285572, A325859, A326359, A326496, A326675, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Divisors[n]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343660(n) + A005361(n).

A063428 a(n) is the smallest positive integer of the form n*k/(n+k).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 6, 5, 10, 3, 12, 7, 6, 8, 16, 6, 18, 4, 12, 11, 22, 6, 20, 13, 18, 12, 28, 5, 30, 16, 22, 17, 10, 9, 36, 19, 26, 8, 40, 6, 42, 22, 18, 23, 46, 12, 42, 25, 34, 26, 52, 18, 30, 7, 38, 29, 58, 10, 60, 31, 14, 32, 40, 22, 66, 34, 46, 20, 70, 8, 72, 37, 30, 38, 28, 26

Offset: 2

Henry Bottomley, Jul 19 2001

nonn

Or, smallest b such that 1/n + 1/c = 1/b has integer solutions.
Largest b is (n-1) since 1/n + 1/(n(n-1)) = 1/(n-1).
a(n) = smallest k such that k*n/(k-n) is an integer. - Derek Orr, May 29 2014

			a(6) = 2 because 6*3/(6+3) = 2 is the smallest integer of the form 6*k/(6+k).
a(10) = 5 since 1/10 + 1/10 = 1/5, 1/10 + 1/15 = 1/6, 1/10 + 1/40 = 1/8, 1/10 + 1/90 = 1/9 and so the first sum provides the value.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A018892, A063427, A127730, A063647, A063648, A063649, A066092.

spi[n_]:=Module[{k=1},While[!IntegerQ[(n*k)/(n+k)],k++];(n*k)/(n+k)]; Array[ spi,80,2] (* Harvey P. Dale, May 05 2022 *)

a(n)={my(k=1); if(n>1, while (n*k%(n + k), k++); n*k/(n + k))} \\ Harry J. Smith, Aug 20 2009

a(n) = n*A063427(n)/(n + A063427(n)) = 2n - A063649(n).

If n is prime a(n) = n - 1. - Benoit Cloitre, Dec 31 2001

New description from Benoit Cloitre, Dec 31 2001
Entry revised by N. J. A. Sloane, Feb 13 2007
Definition revised by Franklin T. Adams-Watters, Aug 07 2009

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

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, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

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

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
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.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

A018894 Numbers k such that sigma(k)/phi(k) sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 1680, 2520, 4620, 9240, 13860, 18480, 27720, 55440, 110880, 120120, 180180, 240240, 360360, 720720, 1441440, 2162160, 3603600, 4084080, 4324320, 6126120, 12252240, 24504480, 36756720, 61261200

Offset: 1

Michel ten Voorde

nonn

Remarkably similar to but ultimately different from A126098. - Jorg Brown and N. J. A. Sloane, Mar 06 2007
Is a(n+1) <= 2*a(n)? Is a(n) divisible by the primorial p# where p is the largest prime divisor of a(n)? Is a(k) divisible by p# for all k > n + 1? (Cf. A002110.) - David A. Corneth, May 22 2016
From Jud McCranie, Nov 28 2017: (Start)
Yes, a(n+1) <= 2*a(n) -- if m is odd, phi(2*m) = phi(m) and sigma(2*m) = 3*sigma(m).
If m is even then phi(2*m) = 2*phi(m) and sigma(2*m) > 2*sigma(m).
So sigma(2*m)/phi(2*m) > sigma(m)/phi(m). (End)
From David A. Corneth, Sep 10 2020: (Start)
Subsequence of A025487.
Let prime(n)# be the product of the first n primes. Then the LCM of the terms <= 10^40 is 89# * 7# * 5# * (3#)^2 * (2#)^4.
We can assume a larger LCM for terms <= 10^60 namely P# * (13#)^3 * (11#) * (5#) * (3#)^2 * (2#)^4. This gives a total of 466 terms <= 10^75 where P is an arbitrary large prime such that P# <= 10^75.
The LCM of these found terms is a proper divisor and for all primes p <= 13 the exponent is less than the assumed prime. Conjecture: These 466 terms are the terms <= 10^75.
For all 240 terms 1 < t <= 10^40 the following holds: there exists a p|t such that t/p is a term. Conjecture: This holds for all terms t > 1.
Using this technique to find terms I get 6522 terms <= 10^1000 and no conflict with terms found above.
See attached file with terms assuming these conjectures. (End)

David A. Corneth, Table of n, a(n) for n = 1..241 (first 79 terms from Jud McCranie)
Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892
David A. Corneth, Conjectured 6522 terms <= 10^1000

Cf. A000010, A000203, A015702, A020492, A025487, A126098, A002110.

Flatten@ Function[k, FirstPosition[k, #] & /@ Union@ Rest@ FoldList[Max, 0, k]]@ Array[DivisorSigma[1, #]/EulerPhi@ # &, 10^7] (* Michael De Vlieger, May 27 2016, Version 10 *)

lista(nn) = {mse = 0; for (n=1, nn, se = sigma(n)/eulerphi(n); if (se > mse, print1(n, ", "); mse = se););} \\ Michel Marcus, Jul 10 2015

More terms from Jud McCranie, Nov 09 2001

Initial term added by Arkadiusz Wesolowski, Sep 06 2012

cp's OEIS Frontend

A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

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A007875 Number of ways of writing n as p*q, with p <= q, gcd(p, q) = 1.

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A015999 a(n) = (tau(n^5) + 4)/5.

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A089233 Number of coprime pairs of divisors > 1 of n.

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A320426 Number of nonempty pairwise coprime subsets of {1,...,n}, where a single number is not considered to be pairwise coprime unless it is equal to 1.

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A015996 (tau(n^4) + 3)/4, where tau = A000005.

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A343652 Number of maximal pairwise coprime sets of divisors of n.

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