cp's OEIS Frontend

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

First differs from A343658 at A(4,2) = 5, A343658(4,2) = 6.

As a triangle, T(n,k) = number of divisors of k^(n-k).

Remarkably similar to but ultimately different from A018894. - Jorg Brown and N. J. A. Sloane, Mar 06 2007

This sequence represents "where records occur" for a number of sequences in addition to A018892 including the following: A015995, A015996, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A048691, A048785, A063647, A117677, A144943. - Ray Chandler, Dec 04 2008

Subsequence of A025487. - Ray Chandler, Sep 05 2008

Also record-setting elements of tau(n^2) (just as A002182 gives the record-setting elements of tau(n)). The point is that A018892 is (tau(n^2) + 1)/2. As tau(n^2) is odd, the record-setting elements of A018892 are also the record setting elements of tau(n^2). - Allen Tracht, Jan 20 2009

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}

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

Inverse Mobius transform of A048785. - R. J. Mathar, Feb 09 2011

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

The multiplicative order of a mod m, GCD(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

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