%I A034444
%S A034444 1,2,2,2,2,4,2,2,2,4,2,4,2,4,4,2,2,4,2,4,4,4,2,4,2,4,2,4,2,8,2,2,4,4,4,
%T A034444 4,2,4,4,4,2,8,2,4,4,4,2,4,2,4,4,4,2,4,4,4,4,4,2,8,2,4,4,2,4,8,2,4,4,8,
%U A034444 2,4,2,4,4,4,4,8,2,4,2,4,2,8,4,4,4,4,2,8,4,4,4,4,4,4,2,4,4,4,2,8,2,4,8
%N A034444 ud(n) = number of unitary divisors of n (d such that d divides n, GCD(d,
               n/d)=1).
%C A034444 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.
%C A034444 Also the number of square-free divisors (Labos E., labos(AT)ana.sote.hu).
%C A034444 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
%C A034444 Also shadow transform of pronic numbers A002378.
%C A034444 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
%C A034444 a(n) is also the number of solutions to x^2 - x == 0 (mod n) . - Yuval 
               Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
%C A034444 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]
%D A034444 R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
%H A034444 T. D. Noe, <a href="b034444.txt">Table of n, a(n) for n=1..10000</a>
%H A034444 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Unitarism and infinitarism</
               a>.
%H A034444 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A034444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               UnitaryDivisor.html">Link to a section of The World of Mathematics.</
               a>
%H A034444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               UnitaryDivisorFunction.html">Unitary Divisor Function</a>
%F A034444 ud(n)=2^(number of different primes dividing n, A001221).
%F A034444 Product_{ p | N } (1 + Legendre(1, p) ).
%F A034444 Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley (se16(AT)btinternet.com), 
               Oct 25 2001
%F A034444 a(n)=sumd( d divides n, mu(d)^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002
%F A034444 a(n)=sum( d divides n, tau(d^2)*mu(n/d) ) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 03 2002
%F A034444 Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, 
               Sep 11 2005.
%F A034444 Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 
               11 2005.
%F A034444 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
%F A034444 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]
%F A034444 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]
%e A034444 Unitary divisors of 12 are 1, 3, 4, 12.
%p A034444 with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) 
               od:
%Y A034444 Cf. A048105, A000173.
%Y A034444 Cf. A013928.
%Y A034444 Cf. A000079, A001221.
%Y A034444 Sequence in context: A058263 A048669 A158522 this_sequence A073180 A127973 
               A023157
%Y A034444 Adjacent sequences: A034441 A034442 A034443 this_sequence A034445 A034446 
               A034447
%K A034444 nonn,nice,easy,mult,new
%O A034444 1,2
%A A034444 N. J. A. Sloane (njas(AT)research.att.com).
%E A034444 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000