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A034444
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ud(n) = number of unitary divisors of n (d such that d divides n, GCD(d,n/d)=1).
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139
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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, 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, 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
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OFFSET
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1,2
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COMMENTS
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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 squarefree divisors (Labos E., labos(AT)ana.sote.hu).
Also number of divisors of the squarefree kernel of n: a(n)=A000005(A007947(n)). - Reinhard Zumkeller, 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 the number of squarefree divisors, but the set of unitary divisors of n is not the set of squarefree divisors, e.g. set of unitary divisors of number 20: {1, 4, 5, 20}, set of squarefree divisors of number 20: {1, 2, 5, 10}. [From Jaroslav Krizek, May 04 2009]
Row lengths of the triangles in A077610 and in A206778. [Reinhard Zumkeller, Feb 12 02]
a(n) is also the number of distinct residues of k^phi(n) (mod n), k=0..n-1. [Michel Lagneau, Nov 15 2012]
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
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LINKS
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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, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary_divisor
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FORMULA
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ud(n)=2^(number of different primes dividing n)= 2^A001221(n).
Product_{ p | N } (1 + Legendre(1, p) ).
Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley, Oct 25 2001
a(n)=sum( d divides n, tau(d^2)*mu(n/d) ), Dirichlet convolution of A048691 and A008683. - Benoit Cloitre, 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) = 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)) * zeta'(2))}/(Pi^2)] + O(sqrt(N)). - Jonathan Vos Post, May 08 2005
a(n)=sum_{d divides n} floor(rad(d)/d), where rad is A007947 and floor(rad(n)/n) = A008966(n). [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 13 2009]
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EXAMPLE
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The a(12)=4 unitary divisors of 12 are 1, 3, 4, 12.
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MAPLE
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with(numtheory): for n from 1 to 200 do printf(`%d, `, 2^nops(ifactors(n)[2])) od:
with(numtheory);
# returns the number of unitary divisors of n and a list of them - from N. J. A. Sloane, May 01 2013
f:=proc(n)
local ct, i, t1, ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d, n/d)=1 then ct:=ct+1; ans:=[op(ans), d]; fi;
od:
RETURN([ct, ans]);
end;
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MATHEMATICA
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a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* From Jean-François Alcover, Apr 05 2011 *)
Table[2^PrimeNu[n], {n, 110}] (* From Harvey P. Dale, Jul 14 2011 *)
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PROG
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(PARI) a(n)=1<<omega(n) \\ Charles R Greathouse IV, Feb 11 2011
(Haskell)
a034444 = length . a077610_row -- Reinhard Zumkeller, Feb 12 2012
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CROSSREFS
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Cf. A077610, A048105, A000173, A013928, A000079, A001221, A034448, A047994.
Sequence in context: A058263 A048669 A158522 * A073180 A183095 A127973
Adjacent sequences: A034441 A034442 A034443 * A034445 A034446 A034447
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KEYWORD
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nonn,nice,easy,mult,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, Jun 20 2000
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STATUS
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approved
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