A070919 -id:A070919 - cp's OEIS frontend

A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

1, 3, 3, 5, 3, 9, 3, 7, 5, 9, 3, 15, 3, 9, 9, 9, 3, 15, 3, 15, 9, 9, 3, 21, 5, 9, 7, 15, 3, 27, 3, 11, 9, 9, 9, 25, 3, 9, 9, 21, 3, 27, 3, 15, 15, 9, 3, 27, 5, 15, 9, 15, 3, 21, 9, 21, 9, 9, 3, 45, 3, 9, 15, 13, 9, 27, 3, 15, 9, 27, 3, 35, 3, 9, 15, 15, 9, 27, 3, 27

Offset: 1

David Johnson-Davies

Inverse Moebius transform of A034444: Sum_{d|n} 2^omega(d), where omega(n) = A001221(n) is the number of distinct primes dividing n.
Number of elements in the set {(x,y): x|n, y|n, gcd(x,y)=1}.
Number of elements in the set {(x,y): lcm(x,y)=n}.
Also gives total number of positive integral solutions (x,y), order being taken into account, to the optical or parallel resistor equation 1/x + 1/y = 1/n. Indeed, writing the latter as X*Y=N, with X=x-n, Y=y-n, N=n^2, the one-to-one correspondence between solutions (X, Y) and (x, y) is obvious, so that clearly, the solution pairs (x, y) are tau(N)=tau(n^2) in number. - Lekraj Beedassy, May 31 2002
Number of ordered pairs of positive integers (a,c) such that n^2 - ac = 0. Therefore number of quadratic equations of the form ax^2 + 2nx + c = 0 where a,n,c are positive integers and each equation has two equal (rational) roots, -n/a. (If a and c are positive integers, but, instead, the coefficient of x is odd, it is impossible for the equation to have equal roots.) - Rick L. Shepherd, Jun 19 2005
Problem A1 on the 21st Putnam competition in 1960 (see John Scholes link) asked for the number of pairs of positive integers (x,y) such that xy/(x+y) = n: the answer is a(n); for n = 4, the a(4) = 5 solutions (x,y) are (5,20), (6,12), (8,8), (12,6), (20,5). - Bernard Schott, Feb 12 2023
Numbers k such that a(k)/d(k) is an integer are in A217584 and the corresponding quotients are in A339055. - Bernard Schott, Feb 15 2023

A. M. Gleason et al., The William Lowell Putnam Mathematical Competitions, Problems & Solutions:1938-1960 Soln. to Prob. 1 1960, p. 516, MAA, 1980.
Ross Honsberger, More Mathematical Morsels, Morsel 43, pp. 232-3, DMA No. 10 MAA, 1991.
Loren C. Larson, Problem-Solving Through Problems, Prob. 3.3.7, p. 102, Springer 1983.
Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Algebra, Prob. 9-9 pp. 143 Dover NY, 1988.
D. O. Shklarsky et al., The USSR Olympiad Problem Book, Soln. to Prob. 123, pp. 28, 217-8, Dover NY.
Wacław Sierpiński, Elementary Theory of Numbers, pp. 71-2, Elsevier, North Holland, 1988.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 91.
Charles W. Trigg, Mathematical Quickies, Question 194, pp. 53, 168, Dover, 1985.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
Ovidiu Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples.
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., Vol. 6 (2003), Article 03.1.6.
John Scholes, Problem A1 of 21st Putnam Competition 1960, 2002. [Wayback Machine link]
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
Index to sequences related to Olympiads and other Mathematical competitions.

Cf. A000005, A034444, Mobius transf. of A035116, A048785, A061391, A018805, A002088, A015614, A018892, A063647, A182139.
Partial sums give A061503.
For similar LCM sequences, see A070919, A070920, A070921.
Cf. A005408, A124010.
For the earliest occurrence of 2n-1 see A016017.
Cf. A001620, A082633, A217584, A339055.

a048691 = product . map (a005408 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Jul 12 2012

A048691[n_]:=DivisorSigma[0,n^2] (* Enrique Pérez Herrero, May 30 2010 *)
DivisorSigma[0,Range[80]^2] (* Harvey P. Dale, Apr 08 2015 *)

numlib::tau (n^2)$ n=1..90 // Zerinvary Lajos, May 13 2008

A048691(n)=prod(i=1,#n=factor(n)[,2],n[i]*2+1) /* or, of course, a(n)=numdiv(n^2) */ \\ M. F. Hasler, Dec 30 2007

for(n=1, 100, print1(direuler(p=2, n, (1 - X^2)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Aug 21 2021

[sigma(n**2, 0) for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

a(n) = A000005(A000290(n)).
tau(n^2) = Sum_{d|n} mu(n/d)*tau(d)^2, where mu(n) = A008683(n), cf. A061391.
Multiplicative with a(p^e) = 2e+1. - Vladeta Jovovic, Jul 23 2001
Also a(n) = Sum_{d|n} (tau(d)*moebius(n/d)^2), Dirichlet convolution of A000005 and A008966. - Benoit Cloitre, Sep 08 2002
a(n) = A055205(n) + A000005(n). - Reinhard Zumkeller, Dec 08 2009
Dirichlet g.f.: (zeta(s))^3/zeta(2s). - R. J. Mathar, Feb 11 2011
a(n) = Sum_{d|n} 2^omega(d). Inverse Mobius transform of A034444. - Enrique Pérez Herrero, Apr 14 2012
G.f.: Sum_{k>=1} 2^omega(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018
Sum_{k=1..n} a(k) ~ n*(6/Pi^2)*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

Additional comments from Vladeta Jovovic, Apr 29 2001

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.

A070920 a(n) = Card{ (x,y,z,u) | lcm(x,y,z,u)=n }.

Original entry on oeis.org

1, 15, 15, 65, 15, 225, 15, 175, 65, 225, 15, 975, 15, 225, 225, 369, 15, 975, 15, 975, 225, 225, 15, 2625, 65, 225, 175, 975, 15, 3375, 15, 671, 225, 225, 225, 4225, 15, 225, 225, 2625, 15, 3375, 15, 975, 975, 225, 15, 5535, 65, 975, 225, 975, 15, 2625, 225

Offset: 1

Benoit Cloitre, May 20 2002

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Antti Karttunen, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.

Cf. A000005, A008683, A048691, A070919, A070921, A247516 (Mobius transform).

Join[{1},Table[Product[(k + 1)^4 - k^4, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^4*moebius(n/d)),","))

a(n) = vecprod(apply(x->(x+1)^4-x^4, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

a(n) = Sum_{d|n} A000005(d)^4*A008683(n/d).
Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^4/k^s.
Multiplicative with a(p^e) = (e+1)^4 - e^4. - Amiram Eldar, Sep 03 2023

A070921 a(n) = Card{ (x,y,z,u,v) | lcm(x,y,z,u,v)=n }.

Original entry on oeis.org

1, 31, 31, 211, 31, 961, 31, 781, 211, 961, 31, 6541, 31, 961, 961, 2101, 31, 6541, 31, 6541, 961, 961, 31, 24211, 211, 961, 781, 6541, 31, 29791, 31, 4651, 961, 961, 961, 44521, 31, 961, 961, 24211, 31, 29791, 31, 6541, 6541, 961, 31, 65131, 211, 6541

Offset: 1

Benoit Cloitre, May 20 2002

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Antti Karttunen, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.

Cf. A000005, A008683, A048691, A070919, A070920, A247517 (Mobius transform).

Join[{1},Table[Product[(k + 1)^5 - k^5, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^5*moebius(n/d)),","))

a(n) = vecprod(apply(x->(x+1)^5-x^5, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

a(n) = Sum_{d|n} A000005(d)^5*A008683(n/d).
Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^5/k^s.
Multiplicative with a(p^e) = (e+1)^5 - e^5. - Amiram Eldar, Sep 03 2023

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

Original entry on oeis.org

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

A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 3, 1, 4, 0, 15, 0, 4, 4, 6, 0, 15, 0, 15, 4, 4, 0, 33, 1, 4, 3, 15, 0, 44, 0, 10, 4, 4, 4, 48, 0, 4, 4, 33, 0, 44, 0, 15, 15, 4, 0, 58, 1, 15, 4, 15, 0, 33, 4, 33, 4, 4, 0, 133, 0, 4, 15, 15, 4, 44, 0, 15, 4, 44, 0, 100, 0, 4, 15, 15, 4, 44, 0, 58, 6, 4, 0, 133, 4, 4, 4, 33, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

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

Cf. A048691, A070919, A086222.

for n from 1 to 100 do a[n] := 0:for x from 1 to n do for y from x+1 to n do for z from y+1 to n do if(lcm(x,y,z)=n) then a[n] := a[n]+1:fi:od:od:od:od:seq(a[j],j=1..200); # Sascha Kurz, Sep 22 2003

f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 0; 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));
A086165(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

a(n) = (A070919(n) - 3*A048691(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004

a(n) = A086222(n) - A048691(n). - Ridouane Oudra, Aug 14 2025

More terms from Sascha Kurz, Sep 22 2003

A247513 Number of elements in the set {(x,y,z): 1<=x,y,z<=n, gcd(x,y,z)=1, lcm(x,y,z)=n}.

Original entry on oeis.org

1, 6, 6, 12, 6, 36, 6, 18, 12, 36, 6, 72, 6, 36, 36, 24, 6, 72, 6, 72, 36, 36, 6, 108, 12, 36, 18, 72, 6, 216, 6, 30, 36, 36, 36, 144, 6, 36, 36, 108, 6, 216, 6, 72, 72, 36, 6, 144, 12, 72, 36, 72, 6, 108, 36, 108, 36, 36, 6, 432

Offset: 1

Ovidiu Bagdasar, Sep 18 2014

For given n and k positive integers, let L(n,k) represent the number of ordered k-tuples of positive integers whose GCD is 1 and LCM is n. In this notation, the sequence corresponds to a(n) = L(n,3).

The inverse Mobius transform is apparently in A070919. - R. J. Mathar, May 25 2017

			The triples corresponding to a(2)=6 are (1,1,2), (1,2,1), (2,1,1), (1,2,2), (2,1,2) and (2,2,1).

Antti Karttunen, Table of n, a(n) for n = 1..10000
O. Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific publications of the state university of Novi Pazar, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, No. 2 (2014), pp. 91-100.

L(n,2) produces A034444.

a:= proc(n) local F; F:= ifactors(n)[2];
      mul(6*f[2],f=F)
end proc:
seq(a(n),n=1..40); # Robert Israel, Sep 22 2014

a[n_] := 6^PrimeNu[n] Times @@ FactorInteger[n][[All, 2]];
Array[a, 60] (* Jean-François Alcover, Jul 27 2020 *)
a[1] = 1; a[n_] := Times @@ (6 * Last[#]& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

a(n) = {f = factor(n); 6^omega(n)*prod(k=1, #f~, f[k, 2]); }

For n = p_1^{n_1} p_2^{n_2}...p_r^{n_r} one has
a(n) = Product_{i=1..r} ((n_i+1)^3 - 2*n_i^3 + (n_i-1)^3).
a(n) = 6^omega(n)*Product_{i=1..r} n_i.
a(n) = 6^A001221(n) *A005361(n). - R. J. Mathar, May 25 2017
Multiplicative with a(p^e) = 6*e. - Amiram Eldar, Sep 26 2020

cp's OEIS Frontend

A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

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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}.

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A070920 a(n) = Card{ (x,y,z,u) | lcm(x,y,z,u)=n }.

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A070921 a(n) = Card{ (x,y,z,u,v) | lcm(x,y,z,u,v)=n }.

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

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A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

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