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A123926 Greatest common divisor of sigma_k(n) for all k >= 1. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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)

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Dirichlet's Theorem

EXAMPLE

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.

MATHEMATICA

a[n_] :=  GCD @@ Table[DivisorSigma[k, n] , {k, 0, EulerPhi[n]}]; Table[a[n], {n, 1, 105}] (* Jean-Fran├žois Alcover, May 21 2012 *)

PROG

(PARI) a(n)=my(d=divisors(n), g=#d); for(k=1, eulerphi(n), g=gcd(lift(sum(i=1, #d, Mod(d[i], g)^k)), g); if(g<3, return(g))); g \\ Charles R Greathouse IV, Jun 17 2013

CROSSREFS

Cf. A109974, A000005, A000203, A001157.

Sequence in context: A082065 A082070 A082902 * A082064 A082055 A073812

Adjacent sequences:  A123923 A123924 A123925 * A123927 A123928 A123929

KEYWORD

nice,nonn

AUTHOR

Franklin T. Adams-Watters, Nov 21 2006

STATUS

approved

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Last modified February 24 03:22 EST 2020. Contains 332195 sequences. (Running on oeis4.)