A258777 Number of points of projective spaces on finite fields.

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, 40, 42, 44, 48, 50, 54, 57, 60, 62, 63, 65, 68, 72, 73, 74, 80, 82, 84, 85, 90, 91, 98, 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

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

Union of 1, A090503 and (A246655 + 1).

Subsequence of A211347.

Cf. A001231, A246655.

Sequence in context: A026500 A091213 A211347 * A298004 A039238 A299532

Adjacent sequences: A258774 A258775 A258776 * A258778 A258779 A258780

Keyword

nonn,nice

Author

Matthieu Pluntz, Jun 09 2015

Status

approved