A122835 -id:A122835 - cp's OEIS frontend

A122836 Number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology.

0, 0, 0, 10, 243, 6131, 202503, 9464302, 641960602, 63249658532, 8976900501699, 1816843604787333, 519355528928422629, 207881392866381430470, 115617051961092253351796, 88736269118240819706018342, 93411113411702066083187522063, 134137950093337685116171325021995, 261492535743634369726764132015849219

Offset: 0

Nathan K. McGregor (mcgregnk(AT)ese.wustl.edu), Sep 15 2006

See comments in A122835.

J. Munkres, Topology, Prentice Hall, (2000), p. 76.

a(n) = A000798(n) - A122835(n).

A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
a122835[n_] := (3/4)*(PolyLog[-n, 1/2] + Boole[n == 0]) - 1/2;
a[n_] := A000798[[n + 1]] - a122835[n];
a /@ Range[0, 18] (* Jean-François Alcover, Jan 01 2020, after Michael Somos in A122835 *)

a(13) corrected and a(16)-a(18) by Jean-François Alcover, Jan 01 2020

A178689 Partial sums of A122836 (number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology).

Original entry on oeis.org

0, 0, 0, 10, 253, 6384, 208887, 9673189, 651633791, 63901292323, 9040801794022, 1825884406581355, 521181413335003984, 208402574279716434454, 115825454535371969786250, 88852094572776191675804592

Offset: 0

Jonathan Vos Post, Dec 25 2010

nonn

All listed (the first 16) values are nonprimes. In the underlying sequence, only A122836(5) = 6131 is prime of the listed values.

			a(4) = 0 + 0 + 0 + 10 + 243 = 253 = 11 * 23.

Cf. A000798, A122835, A122836.

a(n) = Sum_{i=0..n} A122836(i) = Sum_{i=0..n} (A000798(i) - A122835(i)) = (Sum_{i=0..n} A000798(i)) - (Sum_{i=0..n} A122835(i)).

a(13)-a(15) corrected by Georg Fischer, Dec 26 2022

cp's OEIS Frontend

A122836 Number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology.

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