%I #277 Jan 28 2024 09:14:28

%S 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,

%T 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,

%U 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

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

%C 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 Also the number of squarefree divisors of n. - _Labos Elemer_

%C Also number of divisors of the squarefree kernel of n: a(n) = A000005(A007947(n)). - _Reinhard Zumkeller_, Jul 19 2002

%C Also shadow transform of pronic numbers A002378.

%C 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 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 a(n) is the number of squarefree divisors of n, but in general the set of unitary divisors of n is not the set of squarefree divisors (compare the rows of A077610 and A206778). - _Jaroslav Krizek_, May 04 2009

%C Row lengths of the triangles in A077610 and in A206778. - _Reinhard Zumkeller_, Feb 12 2012

%C a(n) is also the number of distinct residues of k^phi(n) (mod n), k=0..n-1. - _Michel Lagneau_, Nov 15 2012

%C 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

%C 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

%C Conjecture: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the unitary divisors of n is divisible by the sum of the k-th 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

%C a(n) is the number of one-digit numbers, k, when written in base n such that k and k^2 end in the same digit. - _Matthew Scroggs_, Jun 01 2018

%C Dirichlet convolution of A271102 and A000005. - _Vaclav Kotesovec_, Apr 08 2019

%C Conjecture: Let b(i; n), n > 0, be multiplicative sequences for some fixed integer i >= 0 with b(i; p^e) = (Sum_{k=1..i+1} A164652(i, k) * e^(k-1)) * (i+2) / (i!) for prime p and e > 0. Then we have Dirichlet generating functions: Sum_{n > 0} b(i; n) / n^s = (zeta(s))^(i+2) / zeta((i+2) * s). Examples for i=0 this sequence, for i=1 A226602, and for i=2 A286779. - _Werner Schulte_, Feb 17 2022

%C The smallest integer with 2^m unitary divisors, or equivalently, the smallest integer with 2^m squarefree divisors, is A002110(m). - _Bernard Schott_, Oct 04 2022

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

%H T. D. Noe, <a href="/A034444/b034444.txt">Table of n, a(n) for n = 1..10000</a>

%H O. Bagdasar, <a href="/A048691/a048691.pdf">On some functions involving the lcm and gcd of integer tuples</a>, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.

%H Masum Billal, <a href="https://arxiv.org/abs/1909.07823">Number of Ways To Write as Product of Co-prime Numbers</a>, arXiv:1909.07823 [math.GM], 2019.

%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 49-50.

%H Lorenz Halbeisen, <a href="http://www.iam.fmph.uniba.sk/amuc/_vol-74/_no_2/_halbeisen/halbeisen.html">A number-theoretic conjecture and its implication for set theory</a>, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.

%H Lorenz Halbeisen and Norbert Hungerbuehler, <a href="http://math.berkeley.edu/~halbeis/publications/pdf/seq.pdf">Number theoretic aspects of a combinatorial function</a>, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.

%H Jon Maiga, <a href="http://jonkagstrom.com/articles/upper_bound_of_fibonacci_entry_points.pdf">Upper bound of Fibonacci entry points</a>, 2019.

%H OEIS Wiki, <a href="https://oeis.org/wiki/Shadow_transform">Shadow transform</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = Sum_{d|n} abs(mu(n)) = 2^(number of different primes dividing n) = 2^A001221(n), with mu(n) = A008683(n). [Added Möbius formula. - _Wolfdieter Lang_, Jan 11 2020]

%F a(n) = Product_{ primes p|n } (1 + Legendre(1, p)).

%F Multiplicative with a(p^k)=2 for p prime and k>0. - _Henry Bottomley_, Oct 25 2001

%F a(n) = Sum_{d|n} tau(d^2)*mu(n/d), Dirichlet convolution of A048691 and A008683. - _Benoit Cloitre_, Oct 03 2002

%F Dirichlet generating function: zeta(s)^2/zeta(2s). - _Franklin T. Adams-Watters_, Sep 11 2005

%F Inverse Mobius transform of A008966. - _Franklin T. Adams-Watters_, Sep 11 2005

%F 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]

%F a(n) = Sum_{d|n} floor(rad(d)/d), where rad is A007947 and floor(rad(n)/n) = A008966(n). - _Enrique Pérez Herrero_, Nov 13 2009

%F a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - _Reinhard Zumkeller_, Jul 30 2013

%F G.f.: Sum_{n>0} A008966(n)*x^n/(1-x^n). - _Mircea Merca_, Feb 25 2014

%F a(n) = Sum_{d|n} lambda(d)*mu(d)), where lambda is A008836. - _Enrique Pérez Herrero_, Apr 27 2014

%F a(n) = A277561(A156552(n)). - _Antti Karttunen_, May 29 2017

%F a(n) = A005361(n^2)/A005361(n). - _Velin Yanev_, Jul 26 2017

%F 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

%F a(n) = Sum_{d|n} A001615(d) * A023900(n/d). - _Torlach Rush_, Jan 20 2020

%F Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = 1. - _Amiram Eldar_, May 29 2020

%e a(12) = 4 because the four unitary divisors of 12 are 1, 3, 4, 12, and also because the four squarefree divisors of 12 are 1, 2, 3, 6.

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

%p with(numtheory);

%p # returns the number of unitary divisors of n and a list of them

%p f:=proc(n)

%p local ct,i,t1,ans;

%p ct:=0; ans:=[];

%p t1:=divisors(n);

%p for i from 1 to nops(t1) do

%p d:=t1[i];

%p if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;

%p od:

%p RETURN([ct,ans]);

%p end;

%p # _N. J. A. Sloane_, May 01 2013

%p # alternative Maple program:

%p a:= n-> 2^nops(ifactors(n)[2]):

%p seq(a(n), n=1..105); # _Alois P. Heinz_, Jan 23 2024

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

%t Table[2^PrimeNu[n],{n,110}] (* _Harvey P. Dale_, Jul 14 2011 *)

%o (PARI) a(n)=1<<omega(n) \\ _Charles R Greathouse IV_, Feb 11 2011

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1+X)/(1-X))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 26 2020

%o (Haskell)

%o a034444 = length . a077610_row -- _Reinhard Zumkeller_, Feb 12 2012

%o (Python)

%o from sympy import divisors, gcd

%o def a(n):

%o return sum(1 for d in divisors(n) if gcd(d, n//d)==1)

%o # _Indranil Ghosh_, Apr 16 2017

%o (Python)

%o from sympy import primefactors

%o def a(n): return 2**len(primefactors(n))

%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Apr 16 2017

%o (Scheme) (define (A034444 n) (if (= 1 n) n (* 2 (A034444 (A028234 n))))) ;; _Antti Karttunen_, May 29 2017

%o (Magma) [#[d:d in Divisors(n)|Gcd(d,n div d) eq 1]:n in [1..110]]; // _Marius A. Burtea_, Jan 11 2020

%o (Magma) [&+[Abs(MoebiusMu(d)):d in Divisors(n)]:n in [1..110]]; // _Marius A. Burtea_, Jan 11 2020

%Y Cf. A077610, A048105, A000173, A013928, A000079, A001221, A002110, A034448, A047994, A061142, A277561.

%Y Sum of the k-th powers of the squarefree divisors of n for k=0..10: this sequence (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

%Y Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: this sequence (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).

%Y Cf. A020821 (Dgf at s=2), A177057 (Dgf at s=4).

%K nonn,nice,easy,mult

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Jun 20 2000