A018892 -id:A018892 - cp's OEIS frontend

A086222 a(n) = card{ (x,y,z) | x <= y <= z and lcm(x,y,z) = n }.

1, 3, 3, 6, 3, 13, 3, 10, 6, 13, 3, 30, 3, 13, 13, 15, 3, 30, 3, 30, 13, 13, 3, 54, 6, 13, 10, 30, 3, 71, 3, 21, 13, 13, 13, 73, 3, 13, 13, 54, 3, 71, 3, 30, 30, 13, 3, 85, 6, 30, 13, 30, 3, 54, 13, 54, 13, 13, 3, 178, 3, 13, 30, 28, 13, 71, 3, 30, 13, 71, 3, 135, 3, 13, 30, 30, 13, 71, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

A number of conjectures are possible, many of which should be easy to prove. Examples: (1) If n is a product of two primes then a(n)=13. (2) If n is a square of a prime then a(n)=6. - John W. Layman, Sep 01 2003

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

Cf. A048691, A070919, A018892, A086165, A377304, A008683.

f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + 3 * Times @@ f2 @@@f + 2) / 6; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

A048691(n) = numdiv(n^2);
A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));
A086222(n) = ((A070919(n)+3*A048691(n)+2)/6); \\ Antti Karttunen, May 19 2017, after Jovovic's formula.

a(n) = {my(e = factor(n)[, 2]); (vecprod(apply(x->(x+1)^3-x^3, e)) + 3*vecprod(apply(x->2*x+1, e)) + 2) / 6;} \\ Amiram Eldar, Sep 03 2023

For a prime p, a(p) = 3.
a(n) = (A070919(n) + 3*A048691(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004
a(n) = Sum_{d|n} A377304(d)*A008683(n/d). - Ridouane Oudra, May 22 2025
a(n) = A086165(n) + A048691(n). - Ridouane Oudra, Aug 19 2025

More terms from John W. Layman, Sep 01 2003

A343660 Number of maximal pairwise coprime sets of at least two 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, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 6, 0, 1, 2, 2, 1, 4, 0, 4, 0, 1, 0, 8, 1, 1, 1

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

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

The case of pairs is A089233.
The case with 1's is A343652.
The case with singletons is (also) A343652.
The non-maximal version is A343653.
The non-maximal version with 1's is A343655.
The version for subsets of {2..n} is A343659 (for n > 2).
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A066620 counts pairwise coprime 3-sets of divisors.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A005361, A007359, A007360, A067824, A074206, A225520, A276187, A320426, A325683, A326077, A326359, A326496, A337485, A343654.

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

a(n) = A343652(n) - A005361(n).

A063649 Largest b such that 1/n=1/c+1/b has integer solutions with c>b.

Original entry on oeis.org

3, 4, 6, 6, 10, 8, 12, 12, 15, 12, 21, 14, 21, 24, 24, 18, 30, 20, 36, 30, 33, 24, 42, 30, 39, 36, 44, 30, 55, 32, 48, 44, 51, 60, 63, 38, 57, 52, 72, 42, 78, 44, 66, 72, 69, 48, 84, 56, 75, 68, 78, 54, 90, 80, 105, 76, 87, 60, 110, 62, 93, 112, 96, 90, 110, 68, 102, 92, 120

Offset: 2

Henry Bottomley, Jul 23 2001

Smallest b is (n+1) since 1/n = 1/(n(n+1))+1/(n+1).

			a(10)=15 since 1/10=1/20+1/20=1/30+1/15=1/35+1/14=1/60+1/12=1/110+1/11, but the first sum does not have c>b, leaving the second sum to provide the value.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A018892, A063427, A063428, A063647, A063648.

f:= proc(n) local b;
  for b from 2*n-1 by -1 do
     if n*b mod (b-n) = 0 then return b fi
od
end proc:
map(f, [$2..100]); # Robert Israel, Dec 01 2019

a[n_] := n + SelectFirst[Divisors[n^2] // Reverse, #Jean-François Alcover, Jun 07 2020 *)

a(n) = n*A063648(n)/(A063648(n)-n) = 2n-A063428(n).
From Robert Israel, Dec 01 2019: (Start)
a(n) = n + A063717(n).
a(n) = n + 1 if and only if n is prime. (End)

A063648 Smallest c such that 1/n=1/c+1/b has integer solutions with c>b.

Original entry on oeis.org

6, 12, 12, 30, 15, 56, 24, 36, 30, 132, 28, 182, 42, 40, 48, 306, 45, 380, 45, 70, 66, 552, 56, 150, 78, 108, 77, 870, 66, 992, 96, 132, 102, 84, 84, 1406, 114, 156, 90, 1722, 91, 1892, 132, 120, 138, 2256, 112, 392, 150, 204, 156, 2862, 135, 176, 120, 228, 174

Offset: 2

Henry Bottomley, Jul 23 2001

nonn

Largest c is n(n+1) since 1/n=1/(n(n+1))+1/(n+1)

			a(10)=30 since 1/10=1/20+1/20=1/30+1/15=1/35+1/14=1/60+1/12=1/110+1/11, but the first sum does not have c>b, leaving the second sum to provide the value.

Cf. A018892, A063427, A063428, A063647, A063649.

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

A063665 Number of ways 1/n can be written as 1/x^2 + 1/y^2 with y >= x >= 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jul 25 2001

nonn

Number of ordered pairs (x,y), with n = (x^2)(y^2)/(x^2 + y^2) and y >= x > 0. - Antti Karttunen, Nov 07 2018

			a(90)=1 since 1/90 = 1/10^2 + 1/30^2
a(98)=2 since 1/98 = 1/10^2 + 1/70^2 = 1/14^2 + 1/14^2.
a(14400) = 3 since 1/14400 = 1/130^2 + 1/312^2 = 1/136^2 + 1/255^2 = 1/150^2 + 1/200^2. - _Antti Karttunen_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000161, A025426, A018892, A063664.

A063665(n) = { my(s=0); for(x=1,n,for(y=x,n,if((n*(x*x+y*y)) == (x*x*y*y), s++))); (s); }; \\ Antti Karttunen, Nov 07 2018

A063665(n) = { my(s=0,y); for(x=sqrtint(n),n,my(x2=x*x); if((x2>n)&&issquare((n*x2)/(x2-n),&y)&&(1==denominator(y))&&(y>=x),s++)); (s); }; \\ Antti Karttunen, Nov 07 2018

Definition clarified by Antti Karttunen, Nov 07 2018

A342267 Number of partitions of 1/n into n reciprocals of positive integers.

Original entry on oeis.org

1, 2, 21, 694, 118995, 132891609

Offset: 1

Ilya Gutkovskiy, Mar 07 2021

			a(2) = 2 because we have 1/2 = 1/4 + 1/4 = 1/3 + 1/6.

Cf. A002966, A003167, A004194, A018892, A020327, A020328, A343074, A343214, A369470.

a(6) from Jud McCranie, Sep 02 2021

A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 5, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 5, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 7, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 2, 3

Offset: 1

Lars Blomberg, Feb 14 2022

nonn

By symmetry, if (i, j) is a solution then so is (j, i). When j=i we get n = 3i, corresponding to the solution 1/2 + 1/2 = 1. Therefore, when 3|n, a(n) > 0 and odd, otherwise a(n) >= 0 and even.
For n < 10^6, the largest term is a(720720) = 285, and 483188 terms are 0.
Other record terms: a(1081080) = 369, a(2162160) = 457, a(3243240) = 481, a(4324320) = 533, a(5405400) = 559, a(6126120) = 597. Record terms with index >= 360360 appear to occur at indices that are multiples of 180180. - Chai Wah Wu, Feb 15 2022

			For n = 3: (i, j) = (1, 1), so a(3) = 1. (1/2 + 1/2 = 1)
For n = 18: (i, j) = (3, 8), (6, 6), (8, 3), so a(18) = 3. (3/15 + 8/10 = 1/5 + 4/5 = 1)
For n = 20: (i, j) = (5, 8), (8, 5), so a(20) = 2.
For n = 36: (i, j) = (6, 16), (8, 15), (12, 12), (15, 8), (16, 6), so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A018892, A016017, A051908, A073546, A227610, A238795, A241883, A297895, A303400.

Cf. A351694, A351695 for records.

a(n)={my(x=n^2, y=2*n); sum(i=1,(n-1)/2, x-=2*n; y-=3; if(x%y==0,1,0))}

from sympy.abc import i, j
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A351532(n):
    return sum(1 for d in diop_quadratic(n**2+3*i*j-2*n*(i+j)) if 0 < d[0] < n and 0 < d[1] < n) # Chai Wah Wu, Feb 15 2022

The original equation can be solved for j giving j = (n(n - 2i)) / (2n - 3i). Varying i from 1 to n-1, a(n) is given by the number of integer values of j, 0 < j < n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786

Offset: 1

Gus Wiseman, May 07 2021

nonn

			The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Antidiagonal sums of the array (or row sums of the triangle) A334997.
A000005 counts divisors of n.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
Array version of A334997 has:
- column k = 2 A007425,
- transpose A077592,
- subdiagonal n = k + 1 A163767,
- strict case A343662 (row sums: A337256),
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]

A376168 Irregular triangle read by rows: row n lists all of the integer pairs (a,b) such that 1/a + 1/b = 1/n, sorted by a.

Original entry on oeis.org

2, 2, 3, 6, 4, 4, 6, 3, 4, 12, 6, 6, 12, 4, 5, 20, 6, 12, 8, 8, 12, 6, 20, 5, 6, 30, 10, 10, 30, 6, 7, 42, 8, 24, 9, 18, 10, 15, 12, 12, 15, 10, 18, 9, 24, 8, 42, 7, 8, 56, 14, 14, 56, 8, 9, 72, 10, 40, 12, 24, 16, 16, 24, 12, 40, 10, 72, 9, 10, 90, 12, 36, 18, 18, 36, 12, 90, 10

Offset: 1

Paolo Xausa, Sep 13 2024

			Triangle begins:
  [1] ( 2, 2);
  [2] ( 3, 6),( 4, 4),( 6, 3);
  [3] ( 4,12),( 6, 6),(12, 4);
  [4] ( 5,20),( 6,12),( 8, 8),(12, 6),(20, 5);
  [5] ( 6,30),(10,10),(30, 6);
  [6] ( 7,42),( 8,24),( 9,18),(10,15),(12,12),(15,10),(18,9),(24,8),(42,7);
  [7] ( 8,56),(14,14),(56, 8);
  [8] ( 9,72),(10,40),(12,24),(16,16),(24,12),(40,10),(72,9);
  [9] (10,90),(12,36),(18,18),(36,12),(90,10);
  ...

Paolo Xausa, Table of n, a(n) for n = 1..14340 (rows 1..400 of triangle, flattened).
Wikipedia, Optic equation.

Cf. A018892, A048691 (row lengths/2), A376169 (row sums).

A376168row[n_] := Module[{a, b}, SolveValues[1/a + 1/b == 1/n && a > 0 && b > 0, {a, b}, Integers]];
Array[A376168row, 10]

T(n,1) = T(n,2*A048691(n)) = n + 1.
T(n,A048691(n)) = T(n,A048691(n) + 1) = n*2.
T(n,k) = T(n,2*A048691(n) - k + 1), with 1 <= k <= 2*A048691(n).

A063716 Number of solutions (r,u,s,t), s>=t, in positive integers to the system of equations 1/r+1/u=1/n, 1/s+1/t=1/u.

Original entry on oeis.org

2, 10, 16, 27, 24, 60, 20, 61, 57, 84, 36, 167, 24, 108, 144, 100, 36, 191, 36, 251, 162, 96, 48, 351, 73, 122, 149, 255, 60, 468, 28, 184, 170, 134, 284, 496, 24, 144, 234, 485, 60, 526, 36, 339, 415, 114, 60, 616, 83, 303, 210, 285, 48, 493, 294, 585, 210, 144, 96

Offset: 1

Vladeta Jovovic, Aug 10 2001

nonn

			For n=2 we have 10 solutions (r,u,s,t), s>=t: (3,6,12,12), (3,6,15,10), (3,6,18,9), (3,6,24,8), (3,6,42,7), (4,4,8,8), (4,4,12,6), (4,4,20,5), (6,3,6,6), (6,3,12,4).

Cf. A000005, A048691, A063715, A018892, A048691, A004194, A063520.

a(n) = (tau(n^2) + Sum_{d|n^2} tau((n+d)^2))/2 = (A048691(n)+A063715(n))/2.

cp's OEIS Frontend

A086222 a(n) = card{ (x,y,z) | x <= y <= z and lcm(x,y,z) = n }.

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A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

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A063649 Largest b such that 1/n=1/c+1/b has integer solutions with c>b.

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A063648 Smallest c such that 1/n=1/c+1/b has integer solutions with c>b.

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A063665 Number of ways 1/n can be written as 1/x^2 + 1/y^2 with y >= x >= 1.

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PARI

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A342267 Number of partitions of 1/n into n reciprocals of positive integers.

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A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

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A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

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