

A034444


a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).


189



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

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..n1.  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
a(n) is the number of irreducible fractions y/x that satisfy x*y=n (and gcd(x,y)=1), x and y positive integers.  Luc Rousseau, Jul 09 2017
a(n) is the number of (x,y) lattice points satisfying both x*y=n and (x,y) is visible from (0,0), x and y positive integers.  Luc Rousseau, Jul 10 2017
Conjecture: For any nonnegative integer k and positive integer n, the sum of the kth powers of the unitary divisors of n is divisible by the sum of the kth powers of the odd unitary divisors of n (note that this sequence lists the sum of the 0th powers of the unitary divisors of n).  Ivan N. Ianakiev, Feb 18 2018
a(n) is the number of onedigit numbers, k, when written in base n such that k and k^2 end in the same digit.  Matthew Scroggs, Jun 01 2018


REFERENCES

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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91100.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
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
Index entries for sequences computed from exponents in factorization of n


FORMULA

a(n) = 2^(number of different primes dividing n) = 2^A001221(n).
a(n) = Product_{ primes 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. AdamsWatters, Sep 11 2005
Inverse Mobius transform of A008966.  Franklin T. AdamsWatters, Sep 11 2005
Asymptotically [Finch] the cumulative sum of a(n) = Sum_{n=1..N} a(n) ~ [6*N*(log N)/(Pi^2)] + [6*N*(2*gamma  1  (12/(Pi^2)) * zeta'(2))]/(Pi^2)] + O(sqrt(N)).  Jonathan Vos Post, May 08 2005 [typo corrected by Vaclav Kotesovec, Sep 13 2018]
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/(1x^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
a(n) = A277561(A156552(n)).  Antti Karttunen, May 29 2017
a(n) = A005361(n^2)/A005361(n).  Velin Yanev, Jul 26 2017
L.g.f.: log(Product_{k>=1} (1  mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.  Ilya Gutkovskiy, Jul 30 2018


EXAMPLE

The a(12) = 4, for the 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
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;
# N. J. A. Sloane, May 01 2013


MATHEMATICA

a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* JeanFrançois 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
(Python)
from sympy import divisors, gcd
def a(n):
s=0
for d in divisors(n):
if gcd(d, n/d)==1: s+=1
return s # Indranil Ghosh, Apr 16 2017
(Python)
from sympy import primefactors
def a(n): return 2**len(primefactors(n))
print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, Apr 16 2017
(Scheme) (define (A034444 n) (if (= 1 n) n (* 2 (A034444 (A028234 n))))) ;; Antti Karttunen, May 29 2017


CROSSREFS

Cf. A077610, A048105, A000173, A013928, A000079, A001221, A034448, A047994, A061142, A277561.
Sequence in context: A232398 A048669 A158522 * A318465 A073180 A316398
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



