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A123926
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Greatest common divisor of sigma_k(n) for all k >= 1.
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0
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1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 6, 4, 2, 2, 2, 1, 2, 4, 2, 2, 4, 2, 3, 4, 2, 4, 1, 2, 2, 4, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 12, 2, 2, 2, 1, 4, 4, 2, 6, 4, 4, 2, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 4, 2, 2, 2, 4, 6, 4, 2, 4, 6, 2, 3, 2, 1, 2, 4, 2, 2, 8
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OFFSET
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1,3
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COMMENTS
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Has the property that if gcd(n,m) = 1, then a(n)*a(m) divides a(n*m). First inequality is a(4) = 1, a(5) = 2, but a(20) = 6. It appears that a(n) also always divides sigma_0(n) = tau(n).
Contribution from Matthew Vandermast, Feb 10 2010: (Start)
1. If an integer m does not divide sigma_0(n), m will also not divide sigma_(totient m)(n). Therefore a(n) always divides sigma_0(n) = tau(n).
2. a(n) is even iff sigma_1(n) is even. Cf. A028982, A028983.
3. a(p)=2 for any odd prime p. If n is an odd integer with 2^e divisors, then a(n)=2^e.
4. For any prime p and positive integer m, if p is congruent to 1 mod m, then a(p^(m-1))=m. It follows from Dirichlet's Theorem (see link) that every positive integer appears in the sequence infinitely often. (End)
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LINKS
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Table of n, a(n) for n=1..105.
Eric Weisstein's World of Mathematics, Dirichlet's Theorem [From Matthew Vandermast, Feb 10 2010]
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EXAMPLE
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For n=4, sigma_1(n) = 7, sigma_2(n) = 21, both divisible by 7, but sigma_3(n) = 73, which is not, so a(4) = 1.
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MATHEMATICA
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a[n_] := GCD @@ Table[DivisorSigma[k, n] , {k, 0, EulerPhi[n]}]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 21 2012 *)
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CROSSREFS
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Cf. A109974, A000005, A000203, A001157.
Sequence in context: A082065 A082070 A082902 * A082064 A082055 A073812
Adjacent sequences: A123923 A123924 A123925 * A123927 A123928 A123929
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KEYWORD
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nice,nonn
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AUTHOR
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Franklin T. Adams-Watters, Nov 21 2006
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STATUS
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approved
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