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%I #35 Oct 23 2020 12:26:20
%S 1,3,4,5,6,7,8,9,10,12,13,14,15,17,18,20,21,24,26,28,30,31,32,33,38,
%T 40,42,44,48,50,54,57,60,62,63,65,68,72,73,74,80,82,84,85,90,91,98,
%U 102,104,108,110,114,121,122,126,127,128,129,132,133,138,140,150,152,156,158,164,168,170,174,180,182,183,192,194,198,200
%N Number of points of projective spaces on finite fields.
%C 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.
%C In other words, numbers that are a repunit in at least one base that is a prime power (A246655). - _Peter Munn_, Oct 21 2020
%H Charles R Greathouse IV, <a href="/A258777/b258777.txt">Table of n, a(n) for n = 1..10000</a>
%H Pierre-Emmanuel Caprace, Pierre de la Harpe, <a href="https://arxiv.org/abs/1807.04992">Groups with irreducibly unfaithful subsets for unitary representations</a>, arXiv:1807.04992 [math.GR], 2018.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%e 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.
%t 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_ *)
%o (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
%Y Union of 1, A090503 and (A246655 + 1).
%Y Subsequence of A211347.
%Y Cf. A001231, A246655.
%K nonn,nice
%O 1,2
%A _Matthieu Pluntz_, Jun 09 2015
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