1,1

The Nagell-Ljunnggren 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 (noe(AT)sspectra.com), Oct 18 2006

Sierpinski 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 (noe(AT)sspectra.com), Oct 18 2006

T. D. Noe, Table of n, a(n) for n=1..500

K. S. Brown, Sum of Divisors Equals a Power of 2

W. Sierpinski, Elementary Theory of Numbers, Warszawa 1964, page 165.

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

For n=2 to 20 sigma[a(n)]=m^n with m=2 or m=4. Computed a(n)'s are products of Mersenne primes[A000608]. Is this true for larger n? Validity of a(11) was tested individually.

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 (noe(AT)sspectra.com), Oct 13 2006

Cf. A006532, A020477, A019422, A019423, A019424, A048257, A048258, A000668, A046528.

Sequence in context: A130060 A053942 A053954 this_sequence A079637 A062662 A084727

Adjacent sequences: A063866 A063867 A063868 this_sequence A063870 A063871 A063872

nonn

Labos E. (labos(AT)ana.sote.hu), Aug 27 2001

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