OFFSET
1,2
COMMENTS
List of integers of form (p^(k*n) - 1)/(p^k - 1) = sigma_k(p^(n-1)) = sum of d^k over all divisors d of p^(n-1), for some prime p and some positive integers k and n. The cardinality of the field is p^k and the dimension of the space is n-1.
In other words, numbers that are a repunit in at least one base that is a prime power (A246655). - Peter Munn, Oct 21 2020
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Pierre-Emmanuel Caprace, Pierre de la Harpe, Groups with irreducibly unfaithful subsets for unitary representations, arXiv:1807.04992 [math.GR], 2018.
Eric Weisstein's World of Mathematics, Repunit
EXAMPLE
7 = (2^(1*3) - 1)/(2^1 - 1) so 7 is in the sequence. 10 = (3^(2*2) - 1)/(3^2 - 1) so 10 is in the sequence.
MATHEMATICA
max = 200; Join[{1}, Select[{#, DivisorSigma[Range[Max[1, Log[#, max] // Floor]], #]}& /@ Range[2, max], PrimePowerQ[#[[1]]]&][[All, 2]] // Flatten // Union] // Select[#, # <= max&]& (* Jean-François Alcover, Jun 24 2015 after Giovanni Resta *)
PROG
(PARI) list(lim)=my(v=List([1]), t); lim\=1; if(lim<2, lim=2); for(k=1, logint(lim - 1, 2), for(n=2, logint(lim*(2^k - 1) + 1, 2)\k, forprime(p=2, , t=(p^(k*n) - 1)/(p^k - 1); if(t>lim, break); listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jun 24 2015
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Matthieu Pluntz, Jun 09 2015
STATUS
approved