# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a034444 Showing 1-1 of 1 %I A034444 #277 Jan 28 2024 09:14:28 %S A034444 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 A034444 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 A034444 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 A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1). %C A034444 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 A034444 Also the number of squarefree divisors of n. - _Labos Elemer_ %C A034444 Also number of divisors of the squarefree kernel of n: a(n) = A000005(A007947(n)). - _Reinhard Zumkeller_, Jul 19 2002 %C A034444 Also shadow transform of pronic numbers A002378. %C A034444 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 A034444 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 A034444 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 A034444 Row lengths of the triangles in A077610 and in A206778. - _Reinhard Zumkeller_, Feb 12 2012 %C A034444 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 A034444 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 A034444 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 A034444 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 A034444 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 A034444 Dirichlet convolution of A271102 and A000005. - _Vaclav Kotesovec_, Apr 08 2019 %C A034444 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 A034444 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 A034444 R. K. Guy, Unsolved Problems in Number Theory, Sect. B3. %H A034444 T. D. Noe, Table of n, a(n) for n = 1..10000 %H A034444 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. %H A034444 Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019. %H A034444 Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author] %H A034444 Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 49-50. %H A034444 Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254. %H A034444 Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150. %H A034444 Jon Maiga, Upper bound of Fibonacci entry points, 2019. %H A034444 OEIS Wiki, Shadow transform. %H A034444 N. J. A. Sloane, Transforms. %H A034444 Eric Weisstein's World of Mathematics, Unitary Divisor. %H A034444 Eric Weisstein's World of Mathematics, Unitary Divisor Function. %H A034444 Wikipedia, Unitary divisor. %H A034444 Index entries for sequences computed from exponents in factorization of n %F A034444 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 A034444 a(n) = Product_{ primes p|n } (1 + Legendre(1, p)). %F A034444 Multiplicative with a(p^k)=2 for p prime and k>0. - _Henry Bottomley_, Oct 25 2001 %F A034444 a(n) = Sum_{d|n} tau(d^2)*mu(n/d), Dirichlet convolution of A048691 and A008683. - _Benoit Cloitre_, Oct 03 2002 %F A034444 Dirichlet generating function: zeta(s)^2/zeta(2s). - _Franklin T. Adams-Watters_, Sep 11 2005 %F A034444 Inverse Mobius transform of A008966. - _Franklin T. Adams-Watters_, Sep 11 2005 %F A034444 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 A034444 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 A034444 a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - _Reinhard Zumkeller_, Jul 30 2013 %F A034444 G.f.: Sum_{n>0} A008966(n)*x^n/(1-x^n). - _Mircea Merca_, Feb 25 2014 %F A034444 a(n) = Sum_{d|n} lambda(d)*mu(d)), where lambda is A008836. - _Enrique Pérez Herrero_, Apr 27 2014 %F A034444 a(n) = A277561(A156552(n)). - _Antti Karttunen_, May 29 2017 %F A034444 a(n) = A005361(n^2)/A005361(n). - _Velin Yanev_, Jul 26 2017 %F A034444 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 A034444 a(n) = Sum_{d|n} A001615(d) * A023900(n/d). - _Torlach Rush_, Jan 20 2020 %F A034444 Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = 1. - _Amiram Eldar_, May 29 2020 %e A034444 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 A034444 with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) od: %p A034444 with(numtheory); %p A034444 # returns the number of unitary divisors of n and a list of them %p A034444 f:=proc(n) %p A034444 local ct,i,t1,ans; %p A034444 ct:=0; ans:=[]; %p A034444 t1:=divisors(n); %p A034444 for i from 1 to nops(t1) do %p A034444 d:=t1[i]; %p A034444 if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi; %p A034444 od: %p A034444 RETURN([ct,ans]); %p A034444 end; %p A034444 # _N. J. A. Sloane_, May 01 2013 %p A034444 # alternative Maple program: %p A034444 a:= n-> 2^nops(ifactors(n)[2]): %p A034444 seq(a(n), n=1..105); # _Alois P. Heinz_, Jan 23 2024 %t A034444 a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* _Jean-François Alcover_, Apr 05 2011 *) %t A034444 Table[2^PrimeNu[n],{n,110}] (* _Harvey P. Dale_, Jul 14 2011 *) %o A034444 (PARI) a(n)=1<