1,2

If n = product p_i^a_i, d = product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.

Also the number of square-free divisors (Labos E., labos(AT)ana.sote.hu).

Also number of divisors of the square-free kernel of n: a(n)=A000005(A007947(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 19 2002

Also shadow transform of pronic numbers A002378.

For n>=1 define an n X n (0,1) matrix A by A[i,j] = 1 if lcm(i,j) = n, A[i,j] = 0 if lcm(i,j) <> n for 1 <= i,j <= n . a(n) is the rank of A . - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003

a(n) is also the number of solutions to x^2 - x == 0 (mod n) . - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003

a(n) is also number of square-free divisors, but set of unitary divisors of n is not set of square-free divisors, e.g. set of unitary divisors of number 20: {1, 4, 5, 20}, set of square-free divisors of number 20: {1, 2, 5, 10}. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), May 04 2009]

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.

T. D. Noe, Table of n, a(n) for n=1..10000

S. R. Finch, Unitarism and infinitarism.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Unitary Divisor Function

ud(n)=2^(number of different primes dividing n, A001221).

Product_{ p | N } (1 + Legendre(1, p) ).

Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley (se16(AT)btinternet.com), Oct 25 2001

a(n)=sumd( d divides n, mu(d)^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002

a(n)=sum( d divides n, tau(d^2)*mu(n/d) ) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 03 2002

Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005.

Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 11 2005.

The number of unitary divisors of an integer n is a(n) = 2^(the number of distinct prime divisors of n) = 2^(smallomega(n)) = 2^A001221(n) = A000079(A001221(n)). Asymptotically [Finch] the cumulative sum of a(n) = SUM[from n=1 to N]a(n) ~ [6*N*(ln N)/(pi^2)] + [6*n*(2*gamma - 1 - (12/(pi^2))*(DerivativeOfReimannZetaFunction(2)))}/(pi^2)] + O(sqrt(N)). - Jonathan Vos Post (jvospost3(AT)gmail.com), May 08 2005

a(n) = 2^A001221(n), a(1) = 1, a(p) = 2, a(pq) = 4, a(pq...z) = 2^k, a(p^k) = 2, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), May 04 2009]

a(n)=sumd( d divides n, floor(rad(n)/n)), where rad(n) is A007947 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 13 2009]

Unitary divisors of 12 are 1, 3, 4, 12.

with(numtheory): for n from 1 to 200 do printf(`%d, `, 2^nops(ifactors(n)[2])) od:

Cf. A048105, A000173.

Cf. A013928.

Cf. A000079, A001221.

Sequence in context: A058263 A048669 A158522 this_sequence A073180 A127973 A023157

Adjacent sequences: A034441 A034442 A034443 this_sequence A034445 A034446 A034447

nonn,nice,easy,mult,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000