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Search: id:A034444
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| A034444 |
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ud(n) = number of unitary divisors of n (d such that d divides n, GCD(d,n/d)=1). |
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+0
109
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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 square-free divisors (Labos E., labos(AT)ana.sote.hu).
Also number of divisors of the square-free kernel of n: a(n)=A000005(A007947(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), 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
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
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LINKS
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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, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
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FORMULA
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ud(n)=2^(number of different primes dividing n, A001221).
Product_{ p | N } (1 + Legendre(1, p) ).
Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley (se16(AT)btinternet.com), Oct 25 2001
a(n)=sumd( d divides n, mu(d)^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002
a(n)=sum( d divides n, tau(d^2)*mu(n/d) ) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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) = 2^(smallomega(n)) = 2^A001221(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))*(DerivativeOfReimannZetaFunction(2)))}/(pi^2)] + O(sqrt(N)). - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 08 2005
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EXAMPLE
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Unitary divisors of 12 are 1, 3, 4, 12.
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MAPLE
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with(numtheory): for n from 1 to 200 do printf(`%d, `, 2^nops(ifactors(n)[2])) od:
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CROSSREFS
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Cf. A048105, A000173.
Cf. A013928.
Cf. A000079, A001221.
Sequence in context: A053238 A058263 A048669 this_sequence A073180 A127973 A023157
Adjacent sequences: A034441 A034442 A034443 this_sequence A034445 A034446 A034447
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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njas
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000
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