A118076 -id:A118076 - cp's OEIS frontend

A066292 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>1.

1, 84, 156, 364, 1092, 435708, 986076, 1118480, 1441188, 1674036, 2446668, 2597868, 3108924, 3875508, 4150692, 5537196, 6066396, 6686316, 13729212, 14639436, 18735444, 23307732, 27092052, 31806684, 58266468, 69728724

Offset: 1

Labos Elemer, Dec 12 2001

nonn

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 < 158. Hence it is easy to verify divisibility for all k. - T. D. Noe, Apr 11 2006

			n=84 is here because 84 divides each one of sigma_4(n)=53771172, sigma_8(n)=2488859101224132, sigma_16(n)=6144339637187846520573009496452, etc.

Cf. A066135, A066284, A066289-A066292.

Cf. A118076.

t={}; Do[If[Mod[DivisorSigma[4,n],n]==0, AppendTo[t,n]], {n,10^8}]; Do[t=Select[t,Mod[DivisorSigma[2^k,# ],# ]==0&],{k,3,20}]; t (* T. D. Noe, Apr 11 2006 *)

Edited by T. D. Noe, Apr 11 2006

A118107 Period of the vector sequence d(n)^2^k mod n for k=1,2,3,..., where d(n) is the vector of divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 6, 2, 1, 1, 2, 1, 4, 2, 10, 1, 1, 1, 4, 1, 2, 1, 6, 4, 2, 6, 3, 1, 1, 1, 4, 2, 1, 1, 4, 1, 1, 10, 2, 1, 2, 1, 6, 4, 6, 4, 2, 1, 1, 1, 4, 1, 2, 1, 3, 3, 4, 1, 2, 2, 10, 4, 11, 6, 1, 1, 6, 4, 4

Offset: 1

T. D. Noe, Apr 13 2006

nonn

This sequence is related to the period of sigma_(2^k)(n) mod n, which is important in studying the numbers n dividing sigma_(2^k)(n) for all k>0. See A066292 and A118076. Note that a(n)=1 if n is a power of a prime.

			See A118106 for an example involving d(n)^k.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A118106 (period of the vector sequence d(n)^k mod n).

Table[d=Divisors[n]; k=0; found=False; While[i=0; While[i

A118107(n) = { my(divs=apply(d -> (d%n),divisors(n)), odivs = Vec(divs), vs = Map()); mapput(vs, odivs, 0); for(k=1,oo,divs = vector(#divs,i,(divs[i]*divs[i])%n); if(mapisdefined(vs, divs), return(k-mapget(vs, divs)), mapput(vs, divs, k))); }; \\ Antti Karttunen, Sep 23 2018

cp's OEIS Frontend

A066292 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>1.

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