

A063869


Least k such that sigma(k)=m^n for some m>1.


3



2, 3, 7, 217, 21, 2667, 93, 217, 381, 651, 2752491, 2667, 8191, 11811, 24573, 57337, 82677, 172011, 393213, 761763, 1572861, 2752491, 5332341, 11010027, 21845397, 48758691, 85327221, 199753347, 341310837, 677207307, 1398273429, 3220807683
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OFFSET

1,1


COMMENTS

For n=2 to 20 sigma(a(n)) = m^n with m=2 or m=4. Computed terms are products of Mersenne primes (A000608). Is this true for larger n? Validity of a(11) was tested individually.
The NagellLjunggren conjecture implies that sigma(x) is never 3^n for n>1. If this is true, then m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2.  T. D. Noe, Oct 18 2006
Sierpiński says that he proved sigma(x) is never 3^r for r>1. Hence m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2.  T. D. Noe, Oct 18 2006


LINKS

T. D. Noe, Table of n, a(n) for n=1..500
K. S. Brown, Sum of Divisors Equals a Power of 2
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 165.


FORMULA

a(n) = Min{x : A000203(x)=m^n} for some m.


MATHEMATICA

Table[ Select[ 1, 800000 ], IntegerQ[ DivisorSigma[ 1, # ]^(1/k) ]& ], {k, 1, 16} ] For n=4, 6, 11, sigma[ a(n) ]=4^11.
d={2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253}; nn=3700; t=Table[Infinity, {nn}]; t[[1]]=2; u={0}; k=1; While[2+d[[k]]<=nn, mer=2^d[[k]]1; Do[a=u[[i]]+d[[k]]; If[a<=nn, If[u[[i]]==0, t[[a]]=Min[t[[a]], mer], t[[a]]=Min[t[[a]], t[[u[[i]]]]*mer]]], {i, Length[u]}]; u=Union[u, u+d[[k]]]; k++ ]; Do[If[t[[i]]==Infinity, t[[i]]=t[[2i]]], {i, nn}]; t (* T. D. Noe, Oct 13 2006 *)


CROSSREFS

Cf. A006532, A020477, A019422, A019423, A019424, A048257, A048258, A000668, A046528.
Sequence in context: A266269 A053942 A053954 * A079637 A062662 A084727
Adjacent sequences: A063866 A063867 A063868 * A063870 A063871 A063872


KEYWORD

nonn


AUTHOR

Labos Elemer, Aug 27 2001


EXTENSIONS

a(24) corrected by T. D. Noe, Oct 15 2006


STATUS

approved



