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A034444 ud(n) = number of unitary divisors of n (d such that d divides n, GCD(d,n/d)=1). 145
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 Elemer

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}. - 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

a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013

REFERENCES

O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, vol. 1, 1 2014.

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

LINKS

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

FORMULA

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). - Enrique Pérez Herrero, Nov 13 2009

G.f.: sum(n>0,A008966(n)*x^n/(1-x^n)). - Mircea Merca, Feb 25 2014

a(n)=sum_{d divides n} lambda(d)*mu(d)), where lambda is A008836. - Enrique Pérez Herrero, Apr 27 2014

EXAMPLE

The a(12)=4 unitary divisors of 12 are 1, 3, 4, 12.

MAPLE

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;

MATHEMATICA

a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* Jean-François Alcover, Apr 05 2011 *)

Table[2^PrimeNu[n], {n, 110}] (* Harvey P. Dale, Jul 14 2011 *)

PROG

(PARI) a(n)=1<<omega(n) \\ Charles R Greathouse IV, Feb 11 2011

(Haskell)

a034444 = length . a077610_row  -- Reinhard Zumkeller, Feb 12 2012

CROSSREFS

Cf. A077610, A048105, A000173, A013928, A000079, A001221, A034448, A047994.

Sequence in context: A232398 A048669 A158522 * A073180 A183095 A242802

Adjacent sequences:  A034441 A034442 A034443 * A034445 A034446 A034447

KEYWORD

nonn,nice,easy,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Jun 20 2000

STATUS

approved

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Last modified October 2 09:45 EDT 2014. Contains 247538 sequences.