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 A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1). 209
 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 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 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 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 Dirichlet convolution of A271102 and A000005. - Vaclav Kotesovec, Apr 08 2019 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), 91--100. Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019. Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author] Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254. Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150. Jon Maiga, Upper bound of Fibonacci entry points, 2019. OEIS Wiki, Shadow transform. 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 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|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 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|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|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))) 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 (* Jean-François Alcover, Apr 05 2011 *) Table[2^PrimeNu[n], {n, 110}] (* Harvey P. Dale, Jul 14 2011 *) PROG (PARI) a(n)=1<

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Last modified October 19 11:09 EDT 2019. Contains 328216 sequences. (Running on oeis4.)