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A118076
Numbers n such that n divides sigma_(2^k)(n), the sum of the 2^k powers of the divisors of n, for all k>0.
2
1, 84, 435708, 986076, 1441188, 6066396, 18735444, 78863148
OFFSET
1,2
COMMENTS
Although these numbers have been tested up to k=20, it is conjectured that n divides sigma_(2^k)(n) for all k>0. Intersection of A046762 and A066292.
Let d be the vector of divisors of n. The sequence d^(2^k) mod n has some period p. Thus if n divides sigma_(2^k)(n) for one period, then n divides sigma_(2^k)(n) for all k. For these n, the first period ends for k<14. Hence it is easy to verify divisibility for all k. Intersection of A046762 and A066292. - T. D. Noe, Apr 12 2006
EXAMPLE
n=84 is here because 84 divides each one of sigma_4(n)=53771172, sigma_8(n)=2488859101224132, sigma_16(n)=6144339637187846520573009496452, etc.
MATHEMATICA
t={}; Do[If[Mod[DivisorSigma[2, n], n]==0, AppendTo[t, n]], {n, 10^8}]; Do[t=Select[t, Mod[DivisorSigma[2^k, # ], # ]==0&], {k, 2, 20}]; t
CROSSREFS
Cf. A076230 (n divides sigma_2(n) and sigma_4(n)).
Sequence in context: A202580 A292196 A076230 * A056746 A111194 A096132
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 11 2006
STATUS
approved