

A078538


Smallest k > 6 such that sigma_n(k)/phi(k) is an integer.


4



12, 22, 12, 249, 12, 22, 12, 19689, 12, 22, 12, 249, 12, 22, 12
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OFFSET

1,1


COMMENTS

For n = 16, 48, 64, and 80 the solutions are hard to find, exceed 10^6 or even 10^7.
If a(16) exists, it is greater than 2^32. Terms a(17) to a(47) are 12, 22, 12, 249, 12, 22, 12, 9897, 12, 22, 12, 249, 12, 22, 12, 2566, 12, 22, 12, 249, 12, 22, 12, 19689, 12, 22, 12, 249, 12, 22, 12.  T. D. Noe, Dec 08 2013


LINKS

Table of n, a(n) for n=1..15.


EXAMPLE

These terms appear as 5th entries in A020492, A015759A015774. k = {1, 2, 3, 6} are solutions to Min{k : Mod[sigma[n, k], phi[k]]=0}. First nontrivial solutions are larger: for odd n, k = 12 is solution; for even powers larger numbers arise like 22, 249, 9897, 19689, etc. Certain powersums of divisors proved to be hard to find.


MATHEMATICA

f[k_, x_] := DivisorSigma[k, x]/EulerPhi[x]; Table[fl=1; Do[s=f[k, n]; If[IntegerQ[s]&&Greater[n, 6], Print[{n, k}; fl=0], {n, 100000}, {k, 1, 100}]


PROG

(PARI) ok(n, k)=my(f=factor(n), r=sigma(f, k)/eulerphi(f)); r>=7 && denominator(r)==1
a(n)=my(k=7); while(!ok(k, n), k++); k \\ Charles R Greathouse IV, Nov 27 2013


CROSSREFS

Cf. A000203, A001157, A001158, A000010, A015759A015774, A020492.
Sequence in context: A212958 A340688 A031186 * A278030 A286094 A098955
Adjacent sequences: A078535 A078536 A078537 * A078539 A078540 A078541


KEYWORD

hard,more,nonn


AUTHOR

Labos Elemer, Nov 29 2002


STATUS

approved



