A063520 -id:A063520 - cp's OEIS frontend

A063123 Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).

1, 2, 3, 3, 4, 4, 4, 6, 6, 4, 6, 6, 4, 8, 10, 5, 6, 6, 6, 12, 8, 4, 8, 12, 6, 8, 12, 6, 8, 8, 6, 12, 8, 8, 18, 9, 4, 8, 16, 8, 8, 8, 6, 18, 12, 4, 10, 15, 9, 12, 12, 6, 8, 16, 16, 16, 8, 4, 12, 12, 4, 12, 21, 14, 16, 8, 6, 12, 16, 8, 12, 12, 4, 12, 18, 12, 16, 8, 10, 25, 10, 4, 12, 24, 8, 8

Offset: 1

Vladeta Jovovic, Aug 08 2001

nonn

Unordered solutions to the equation 1/n = 1/r+1/s+1/(r*s) are r=d+n, s=n*(n+1)/d+n, where d is factor of n*(n+1) not greater than n.

Number of divisors of n-th oblong number not greater than n. - Chandler

			a(2)=2 because 1/2=1/3+1/8+1/24=1/4+1/5+1/20.

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A063520.

a[n_]:=DivisorSigma[0,n]DivisorSigma[0,(n+1)]/2; Array[a,86] (* Stefano Spezia, Aug 11 2025 *)

a(n) = numdiv(n)*numdiv(n+1)/2 \\ Michel Marcus, Jun 17 2013

a(n) = tau(n)*tau(n+1)/2 = A092517(n)/2.

A063525 Sum divides product: number of ordered triples of positive solutions (r,s,t) to the equation rst = n(r+s+t).

Original entry on oeis.org

6, 15, 28, 30, 48, 45, 45, 78, 75, 54, 84, 94, 48, 105, 132, 105, 84, 99, 78, 189, 138, 60, 111, 210, 90, 132, 184, 129, 114, 153, 102, 228, 141, 105, 294, 267, 48, 132, 234, 228, 132, 159, 78, 300, 270, 96, 159, 301, 144, 231, 228, 162, 120, 297, 270, 429, 144, 72

Offset: 1

Jud McCranie Aug 01 2001

nonn

			The ordered solutions (r,s,t) of rst = 3(r+s+t) are (1,4,15), (1,5,9), (1,6,7), (2,2,12), (2,3,5), (3,3,3) for a total of 28 permuted solutions, hence a(3) = 28.

Giovanni Resta, Table of n, a(n) for n = 1..500
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. A063520.

np[{x_,y_,z_}] := If[x==y==z, 1, If[x==y || y==z, 3, 6]]; f[n_, t_] := Block[{s, r, sol = Reduce[r s t == n (r + s + t) && r >= s >= t , {r, s}, Integers]}, If[sol === False, 0, Total[np /@ ({r, s, t} /. List@ ToRules@ sol)]]]; a[n_] := Sum[f[n, t], {t, Sqrt[3 n]}]; Array[a, 58] (* Giovanni Resta, Jun 06 2019 *)

More terms from David W. Wilson, Aug 01 2001

A065015 Sum divides product: number of integer solutions (w,x,y,z), w >= x >= y >= z > 0, to the equation wxyz = n(w+x+y+z).

Original entry on oeis.org

1, 5, 4, 8, 6, 14, 7, 15, 12, 22, 7, 26, 9, 24, 23, 30, 8, 42, 11, 31, 21, 29, 12, 56, 18, 32, 27, 48, 13, 68, 11, 42, 35, 40, 32, 65, 9, 35, 30, 76, 9, 87, 18, 51, 49, 39, 18, 97, 27, 70, 31, 52, 14, 108, 36, 72, 33, 52, 17, 120, 15, 37, 66, 90, 36, 93, 12, 63, 44, 123, 19, 130

Offset: 1

John W. Layman, Nov 01 2001

nonn

See A063520 for the corresponding problem in three variables.

			a(7) = 7, since there are seven such solutions to wxyz = 7(w+x+y+z): (42,2,2,2), (49,4,2,1), (7,4,4,1), (10,7,2,1), (70,8,1,1), (28,10,1,1) and (16,14,1,1).

Cf. A063520, A071690.

A065015(n,d=0)={sum(x=1,sqrtn(4*n+.5,3),sum(y=x,sqrtint(4*n\x),sum(z=max(y,n\(x*y)+1),4*n\(x*y),(x+y+z)*n%(x*y*z-n)==0&&(x+y+z)*n>=(x*y*z-n)*z&&!(d&&print1([x,y,z,t=(x+y+z)*n/(x*y*z-n),x*y*z*t/(x+y+z+t)])))))} \\ M. F. Hasler, Aug 01 2015

A063715 Number of solutions (r,u,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

3, 17, 29, 49, 45, 111, 37, 115, 109, 159, 69, 319, 45, 207, 279, 191, 69, 367, 69, 487, 315, 183, 93, 681, 141, 235, 291, 495, 117, 909, 53, 357, 331, 259, 559, 967, 45, 279, 459, 949, 117, 1025, 69, 663, 815, 219, 117, 1205, 161, 591, 411, 555, 93, 965, 579

Offset: 1

Vladeta Jovovic, Aug 10 2001

nonn

			For n=2 we have 17 solutions (r,u,s,t): (3,6,7,42), (3,6,8,24), (3,6,9,18), (3,6,10,15), (3,6,12,12), (3,6,15,10), (3,6,18,9), (3,6,24,8), (3,6,42,7), (4,4,5,20), (4,4,6,12), (4,4,8,8), (4,4,12,6), (4,4,20,5), (6,3,4,12), (6,3,6,6), (6,3,12,4).

Cf. A000005, A048691, A063716, A048691, A004194, A063520.

a(n) = sumdiv(n^2, d, numdiv((n+d)^2)) \\ Michel Marcus, Jun 17 2013

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

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.

A071694 Number of ways to write n as n = xyz/(x+y+z) 1 <= x <= y <= z <= n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 2, 2, 1, 2, 3, 1, 3, 5, 3, 2, 3, 2, 7, 5, 1, 3, 8, 2, 4, 8, 4, 3, 5, 2, 9, 5, 2, 14, 12, 1, 4, 10, 10, 3, 5, 2, 11, 11, 2, 4, 13, 5, 10, 8, 6, 3, 14, 12, 22, 5, 1, 5, 14, 1, 5, 22, 11, 13, 11, 2, 11, 9, 6, 5, 19, 1, 5, 19, 10, 13, 6, 4, 33, 14, 1, 5, 24, 6, 6, 11, 10, 5

Offset: 1

Benoit Cloitre, Jun 23 2002

nonn

If n is a prime other than 3, then a(n) = A000005(n+1)/2 - 1. - Robert Israel, Oct 29 2018

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A000005, A063520.

f:= proc(n) local x,y,z,t;
t:= 0;
  for x from 1 to n do
    for y from max(x,ceil(n/x)) to n do
      if x*y = n then next fi;
      z:= n*(x+y)/(x*y-n);
      if z::integer and z>=y and z<=n then t:= t+1 fi
od od:
t
end proc:
map(f, [$1..100]); # Robert Israel, Oct 29 2018

a[n_] := Sum[Boole[x y z/(x + y + z) == n], {x, n}, {y, x}, {z, y}];
Array[a, 100] (* Jean-François Alcover, Aug 24 2020 *)

for(n=1,90,print1(sum(a=1,n,sum(b=1,a,sum(c=1,b,if(a*b*c/(a+b+c)-n,0,1)))),","))

A078173 a(n) = max(min(r,s,t)), where the maximum is taken over all positive integer solutions of the Diophantine equation rst = n * (r+s+t).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 3, 4, 4, 4, 4, 6, 3, 5, 5, 6, 4, 6, 5, 6, 7, 4, 5, 6, 5, 7, 9, 7, 6, 6, 5, 8, 9, 6, 8, 8, 3, 8, 9, 10, 7, 7, 5, 10, 9, 7, 7, 12, 7, 10, 9, 9, 7, 10, 11, 12, 9, 6, 7, 10, 7, 8, 11, 12, 13, 11, 5, 11, 9, 10, 9, 12, 5, 8, 15, 10, 12, 12, 9, 12, 12, 7, 8, 14, 15, 11, 11, 12, 10, 15

Offset: 1

John W. Layman, Nov 20 2002

nonn

Conjecture: max(t) = n*(n+2) and max(s) = 2*n, where the maxima are taken over all positive integer solutions r<=s<=t of the above Diophantine equation.

Cf. A063520.

cp's OEIS Frontend

A063123 Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A063525 Sum divides product: number of ordered triples of positive solutions (r,s,t) to the equation rst = n(r+s+t).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A065015 Sum divides product: number of integer solutions (w,x,y,z), w >= x >= y >= z > 0, to the equation w*x*y*z = n*(w+x+y+z).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A063715 Number of solutions (r,u,s,t) in positive integers to the system of equations 1/r + 1/u = 1/n, 1/s + 1/t = 1/u.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Examples

Crossrefs

Formula

A071694 Number of ways to write n as n = x*y*z/(x+y+z) 1 <= x <= y <= z <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A078173 a(n) = max(min(r,s,t)), where the maximum is taken over all positive integer solutions of the Diophantine equation r*s*t = n * (r+s+t).

Views

Author

Keywords

Comments

Crossrefs

A065015 Sum divides product: number of integer solutions (w,x,y,z), w >= x >= y >= z > 0, to the equation wxyz = n(w+x+y+z).

A071694 Number of ways to write n as n = xyz/(x+y+z) 1 <= x <= y <= z <= n.

A078173 a(n) = max(min(r,s,t)), where the maximum is taken over all positive integer solutions of the Diophantine equation rst = n * (r+s+t).