A001231 -id:A001231 - cp's OEIS frontend

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

Matthieu Pluntz, Jun 09 2015

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

			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.

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

Union of 1, A090503 and (A246655 + 1).
Subsequence of A211347.
Cf. A001231, A246655.

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 *)

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

A342307 Table read by ascending antidiagonals: T(n, k) is the maximum number of quasi k-gons that are not k-gons in a finite projective plane of order n, with k >= 3.

Original entry on oeis.org

126, 936, 2520, 3780, 41184, 25200, 11160, 287280, 1029600, 151200, 27090, 1294560, 12927600, 18532800, 529200, 57456, 4442760, 90619200, 439538400, 259459200, 846720, 110376, 12640320, 444276000, 4893436800, 12307075200, 2905943040, 0, 196560, 31346784, 1706443200, 34653528000, 222651374400, 295369804800, 26153487360, 0

Offset: 2

Stefano Spezia, Mar 08 2021

			n\k |     3        4         5           6
----+-------------------------------------
  2 |   126     2520     25200      151200 ...
  3 |   936    41184   1029600    18532800 ...
  4 |  3780   287280  12927600   439538400 ...
  5 | 11160  1294560  90619200  4893436800 ...
...

Vladislav Taranchuk, On the number of k-gons in finite projective planes, (2020).

Cf. A000142, A000217, A000794, A001231, A124278, A124287, A293819.

T[n_,k_]:=k!Binomial[k-1,2]Binomial[n^2+n+1,k-1](n-1); Table[T[n-k+3,k],{n,2,9},{k,3,n+1}]//Flatten

T(n, k) = k!*binomial(k - 1, 2)*binomial(n^2 + n + 1, k - 1)*(n - 1).

cp's OEIS Frontend

A258777 Number of points of projective spaces on finite fields.

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