Hilko Koning - cp's OEIS frontend

A385919 Number of non-isomorphic round-robin tournament schedules for 2*n players, where the order of rounds does not matter.

1, 1, 6, 6930, 12257280, 526915620, 1132835421602062347

Offset: 1

Peter Boonstra and Hilko Koning, Jul 25 2025

A round-robin tournament schedule with 2*n players consists of 2*n-1 rounds, where in each round the players are divided into n disjoint pairs, and every player plays against every other player exactly once.
Also the number of non-isomorphic 1-factorizations of the complete graph K_{2n}. We count 1-factorizations of the complete graph K_{2n} up to isomorphism, where 'isomorphism' means that two factorizations are considered the same if one can be transformed into the other by:
(1) relabeling the vertices (i.e., permuting the players), and
(2) reordering the rounds (i.e., permuting the 1-factors).
This is equivalent to counting round-robin tournament schedules where players are unlabeled and the order of the rounds is irrelevant.
Number of ways to partition the edge set of K_{2n} into 2n-1 perfect matchings (1-factors), up to isomorphism. Also the number of non-isomorphic 1-factorizations of the complete graph K_{2n}.

			a(1) = 1: one match between two players.
a(2) = 1: three matches (A-B, C-D, etc) organized into three rounds. All factorizations are isomorphic.
a(3) = 6: The 15 edges of K_6 can be partitioned into 5 rounds of 3 matches in 6 non-isomorphic ways.

Colbourn and Dinitz, CRC Handbook of Combinatorial Designs, 2nd ed. (2006), entry on 1 factorizations of complete graphs.

Petteri Kaski and Patric R. J. Östergård, There are 1132835421602062347 nonisomorphic one-factorizations of K_14, arXiv:0801.0202 [math.CO], 2007.
Hilko Koning, Non-isomorphic 1-factorizations of K_6 (6 labeled players, 5 rounds each).
Brendan D. McKay and Ian M. Wanless, There are 526915620 nonisomorphic one-factorizations of K_12.
Wikipedia, 1-factorization.
R. M. Wilson, Decompositions of complete graphs into subgraphs isomorphic to a given graph, Congressus Numerantium 15 (1976), pp. 647-659.

Cf. A000085 (number of involutive permutations), A000569 (number of 1-factorizations of K_{2n}, not up to isomorphism).

(* n=4 is extremely memory- and CPU-intensive. The Mathematica approach is theoretically correct but utterly infeasible for n >= 4 *)
ClearAll[nonIsomorphic1Factorizations];
nonIsomorphic1Factorizations[n_Integer?Positive] :=
  Module[{vertices = Range[2 n], edges, matchings, factorizations,
    perms, canonical, relabel, isIsomorphicQ, nonIsomorphicList = {}},
    edges = Subsets[vertices, {2}];
   matchings =
    Select[Subsets[edges, {n}], DuplicateFreeQ[Flatten[#]] &];
   factorizations =
    Select[Subsets[matchings, {2 n - 1}], DuplicateFreeQ[Join @@ #] &];
   canonical[fact_] := Sort[Sort /@ fact];
   perms = Permutations[vertices];
   relabel[fact_, perm_] :=
    Sort[Sort /@ (Sort /@
          Replace[#, {a_, b_} :>
            Sort[{perm[[a]], perm[[b]]}], {2}] & /@ fact)];
   isIsomorphicQ[f1_, f2_] :=
    MemberQ[relabel[f1, #] & /@ perms, canonical[f2]];
   Do[If[NoneTrue[nonIsomorphicList, isIsomorphicQ[fact, #] &],
     AppendTo[nonIsomorphicList, fact]], {fact, factorizations}];
   nonIsomorphicList];
(*Display the number of non-isomorphic 1-factorizations for K_{2n} for n=1 to 5*)
Table[With[{list = nonIsomorphic1Factorizations[n]},
  Print["n = ", n, " \[RightArrow] ", Length[list],
   " non-isomorphic 1-factorizations of K_", 2 n];
  Length[list]], {n, 1, 5}]

A317626 Intersections with the x-axis of a bouncing ball on a Sophie Germain billiard table.

Original entry on oeis.org

2, 4, 8, 10, 14, 18, 28, 30, 38, 44, 58, 60, 64, 78, 80, 84, 94, 98, 120, 140, 144, 148, 164, 170, 198, 214, 218, 220, 228, 240, 248, 254, 270, 304, 318, 338, 340, 344, 350, 368, 408, 410, 430, 470, 480, 484, 494, 500, 504, 520, 528, 534, 578, 604, 630, 634, 644, 658

Offset: 1

Hilko Koning, Aug 02 2018

nonn

In the first quadrant of a coordinate system define a rectangular Sophie Germain billiard table with width p and length 2p+1, with vertices (0,0), (p,0), (p,2p+1) and (0,2p+1). A billiard ball (considered to be a point) starts from (0,0) at an angle of 45 degrees and hits the sides exactly p times until it hits the x-axis. The sequence gives the intersections with the x-axis of consecutive Sophie Germain prime numbers (p > 3) after p bounces.
The sum of all crossed lattice points (including the rectangle sides) is the sum of crossed points left under, right middle and left up respectively ((p+7)/6)^2 + (p+1)(p+4)/18 + (p+1)(p+7)/36 = ((p+4)/3)^2 (see bouncing examples).
The enclosed areas in the Sophie Germain billiard table also correspond to ((p+4)/3)^2.
The number of trajectories is a subsequence of A176045.
The number of trajectories with slope +1 or with slope -1 is a subsequence of A124485.
The sum of a term of this sequence and the corresponding Sophie Germain prime is A317510 and it appears that this is a subsequence of A179882. Checked up to and including 33295 of A317510 (Sophie Germain prime 24971).

Samuel King, Billiard Simulator
Hilko Koning, Some bouncing examples.

Cf. A005384, A151922, A176045, A124485, A179882, A317510.

a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018

lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (p + 1)/3]], {p, 5, 2*10^3}]; lst
(Select[Prime@ Range[3, 300], PrimeQ[2# + 1] &] + 1)/3 (* Robert G. Wilson v, Aug 02 2018 *)

lista(nn) = forprime(p=2, nn, if (isprime(2*p+1), print1((p+1)/3, ", "));); \\ Michel Marcus, Aug 25 2018

a(n) = (A005384(n)+1)/3 for n>=3. - Michel Marcus, Aug 25 2018

A317510 Numbers (4p+1)/3 where p is a Sophie Germain prime p > 3.

Original entry on oeis.org

7, 15, 31, 39, 55, 71, 111, 119, 151, 175, 231, 239, 255, 311, 319, 335, 375, 391, 479, 559, 575, 591, 655, 679, 791, 855, 871, 879, 911, 959, 991, 1015, 1079, 1215, 1271, 1351, 1359, 1375, 1399, 1471, 1631, 1639, 1719, 1879, 1919, 1935, 1975, 1999, 2015, 2079, 2111, 2135, 2311, 2415, 2519, 2535, 2575, 2631

Offset: 1

Hilko Koning, Jul 30 2018

nonn

It appears that this is a subsequence of A179882.
Define a set of consecutive positive odd numbers {1,......, (A077065(n)-1)} with n >= 3 and skip the number A077065(n)/2. Then the contraharmonic mean of that set gives the sequence. For example: ContraharmonicMean[{1, 3, 7, 9}] = 7, ContraharmonicMean[{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}] = 15, ContraharmonicMean[{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37,39, 41, 43, 45}] = 31. - Hilko Koning, Aug 28 2018
Let p be a Sophie Germain prime and define h = 2p + 1 a safe prime. Then the contraharmonic mean of the totatives of h is given by: CHM(h) =  (Sum_{1 <= k < h, gcd(k, h) = 1} k^2) / (Sum_{1 <= k < h, gcd(k, h) = 1} k). Since h is prime, all integers k = 1, 2, ... , h - 1 are coprime to h. Then, CHM(h) = ((h - 1) * h * (2h-1) / 6) / ((h - 1) * h  / 2). Thus CHM = (2h-1) / 3 = (4p+1) / 3. These values are integers precisely when p == 2 mod 3, which holds for all Sophie Germain primes, p >= 5. The resulting values for the sequence A317510, which is therefore a subsequence of A179882. - Hilko Koning, Jun 17 2025

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A005384, A179882.

Subsequence of A004767, and of A004771.

a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(4*p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018

lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (4 p + 1)/3]], {p, 5, 2*10^3}]; lst
4 (Select[Prime@Range[3, 300], PrimeQ[2 # + 1] &] + 1)/3 - 1 (* Robert G. Wilson v, Jul 30 2018 *)

lista(nn) = {forprime (p=5, nn, if (isprime(2*p+1), print1((4*p+1)/3, ", ")););} \\ Michel Marcus, Aug 27 2018

A275770 Primes p == 5 (mod 6) that are not Sophie Germain primes.

Original entry on oeis.org

17, 47, 59, 71, 101, 107, 137, 149, 167, 197, 227, 257, 263, 269, 311, 317, 347, 353, 383, 389, 401, 449, 461, 467, 479, 503, 521, 557, 563, 569, 587, 599, 617, 647, 677, 701, 773, 797, 821, 827, 839, 857, 863, 881, 887, 929, 941, 947, 971, 977, 983, 1061, 1091, 1097

Offset: 1

Hilko Koning, Aug 08 2016

nonn

Cf. A005384, A007528, A045979, A053176, A156543.

Select[Prime[Range[150]], Mod[#, 6] == 5 && \[Not] PrimeQ[2 # + 1] &]

forprime(p=1, 900, if(Mod(p, 6)==5 && !ispseudoprime(2*p+1), print1(p, ", "))) \\ Felix Fröhlich, Aug 08 2016

Subset of A156543 and subset of A045979

Intersection of A007528 and A053176. - Felix Fröhlich, Aug 08 2016

A095946 Primes of the form 8n^4 - 2n^3 - 132n^2 + 318n + 80557.

Original entry on oeis.org

80557, 80749, 80777, 80917, 81637, 83597, 87649, 94837, 106397, 123757, 148537, 182549, 227797, 286477, 360977, 453877, 567949, 706157, 871657, 1067797, 1298117, 1566349, 1876417, 2232437, 2638717, 3099757, 3620249, 4205077, 4859317

Offset: 1

Hilko Koning (hilko(AT)hilko.net), Jul 13 2004

The first composite has n = 29.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein, Prime-Generating Polynomials

Cf. A076808, A076809.

[a: n in [0..100] | IsPrime(a) where a is 8*n^4-2*n^3-132*n^2+ 318*n+80557]; // Vincenzo Librandi, Jul 17 2012

Select[Table[8n^4-2n^3-132n^2+318n+80557,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Jul 17 2012 *)

A095974 Evaluate n^4 - 93n^3 + 3196n^2 - 48008n + 265483 for n >= 0, record the primes.

Original entry on oeis.org

265483, 220579, 181523, 147793, 118891, 94343, 73699, 56533, 42443, 31051, 22003, 14969, 9643, 5743, 3011, 1213, 139, -397, -557, -479, -277, -41, 163, 293, 331, 283, 179, 73, 43, 191, 643, 1549, 3083, 5443

Offset: 0

Hilko Koning (hilko(AT)hilko.net), Jul 16 2004

A prime-generating quartic polynomial.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A076808, A076809, A095946.

A096372 List of primes produced by a certain "prime-generating" quartic polynomial.

Original entry on oeis.org

1688311, 1410743, 1168619, 958807, 778319, 624311, 494083, 385079, 294887, 221239, 162011, 115223, 79039, 51767, 31859, 17911, 8663, 2999, -53, -1321, -1489, -1097, -541, -73, 199, 311, 443, 919, 2207, 4919, 9811, 17783, 29879, 47287, 71339, 103511, 145423, 198839

Offset: 1

Hilko Koning (hilko(AT)hilko.net), Jul 19 2004

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A076808, A076809, A095946, A095974.

f:= n -> 6*n^4 - 558*n^3 + 19354*n^2 - 296370*n + 1688311:
select(isprime@abs, [seq(f(n),n=0..100)]); # Robert Israel, Jan 16 2018

Select[Table[6n^4-558n^3+19354n^2-296370n+1688311,{n,0,40}],PrimeQ] (* Harvey P. Dale, Dec 31 2018 *)

Evaluate 6n^4 - 558n^3 + 19354n^2 - 296370n + 1688311 for n >= 0, record the primes.

Offset changed to 1 by Robert Israel, Jan 16 2018

A076808 a(n) = 82n^3 - 1228n^2 + 6130n - 5861.

Original entry on oeis.org

-5861, -877, 2143, 3691, 4259, 4339, 4423, 5003, 6571, 9619, 14639, 22123, 32563, 46451, 64279, 86539, 113723, 146323, 184831, 229739, 281539, 340723, 407783, 483211, 567499, 661139, 764623, 878443, 1003091, 1139059, 1286839, 1446923, 1619803, 1805971

Offset: 0

Hilko Koning (hilko(AT)hilko.net), Nov 18 2002

A prime-generating cubic polynomial.

For n=0 ... 31, the absolute value of terms in this sequence are primes. This is not the case for n=32. See A272323 and A272324. - Robert Price, Apr 25 2016

Eric Weisstein's World of Mathematics, Prime-Generating Polynomials
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A050266, A076809, A272323, A272324.

Table[82 n^3 - 1228 n^2 + 6130 n - 5861, {n, 0, 31}] (* or *)
CoefficientList[Series[(13301 x^3 - 29515 x^2 + 22567 x - 5861)/(x - 1)^4, {x, 0, 31}], x] (* Michael De Vlieger, Apr 25 2016 *)
LinearRecurrence[{4,-6,4,-1},{-5861,-877,2143,3691},40] (* Harvey P. Dale, Jun 18 2018 *)

A076808(n):=82*n^3-1228*n^2+6130*n-5861$
makelist(A076808(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

a(n)=82*n^3-1228*n^2+6130*n-5861 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (13301*x^3-29515*x^2+22567*x-5861)/(x-1)^4. - Colin Barker, Nov 10 2012

E.g.f.: (-5861 + 4984*x - 982*x^2 + 82*x^3)*exp(x). - Ilya Gutkovskiy, Apr 25 2016

A076809 a(n) = n^4 + 853n^3 + 2636n^2 + 3536n + 1753.

Original entry on oeis.org

1753, 8779, 26209, 59197, 112921, 192583, 303409, 450649, 639577, 875491, 1163713, 1509589, 1918489, 2395807, 2946961, 3577393, 4292569, 5097979, 5999137, 7001581, 8110873, 9332599, 10672369, 12135817, 13728601, 15456403, 17324929, 19339909, 21507097, 23832271, 26321233, 28979809, 31813849, 34829227

Offset: 0

Hilko Koning (hilko(AT)hilko.net), Nov 18 2002

A prime-generating quartic polynomial.

For n=0 ... 20, the terms in this sequence are primes. This is not the case for n=21. See A272325 and A272326. - Robert Price, Apr 25 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A076808, A272325, A272326.

A076809:=n->n^4 + 853*n^3 + 2636*n^2 + 3536*n + 1753; seq(A076809(n), n=0..100); # Wesley Ivan Hurt, Nov 13 2013

Table[n^4 + 853n^3 + 2636n^2 + 3536n + 1753, {n,0,100}] (* Wesley Ivan Hurt, Nov 13 2013 *)
CoefficientList[Series[-(x^4 - 1588 x^3 - 156 x^2 + 14 x + 1753)/(x - 1)^5, {x, 0, 33}], x] (* Michael De Vlieger, Apr 25 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{1753,8779,26209,59197,112921},40] (* Harvey P. Dale, Jan 20 2025 *)

A076809(n):=n^4 + 853*n^3 + 2636*n^2 + 3536*n + 1753$
makelist(A076809(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

G.f.: -(x^4-1588*x^3-156*x^2+14*x+1753)/(x- 1)^5. [Colin Barker, Nov 11 2012]

E.g.f.: (1753 + 7026*x + 5202*x^2 + 859*x^3 + x^4)*exp(x). - Ilya Gutkovskiy, Apr 25 2016

More terms from Michael De Vlieger, Apr 25 2016

cp's OEIS Frontend

User: Hilko Koning

A385919 Number of non-isomorphic round-robin tournament schedules for 2*n players, where the order of rounds does not matter.

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A317626 Intersections with the x-axis of a bouncing ball on a Sophie Germain billiard table.

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A317510 Numbers (4p+1)/3 where p is a Sophie Germain prime p > 3.

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GAP

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A275770 Primes p == 5 (mod 6) that are not Sophie Germain primes.

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A095946 Primes of the form 8n^4 - 2n^3 - 132n^2 + 318n + 80557.

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Magma

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A095974 Evaluate n^4 - 93n^3 + 3196n^2 - 48008n + 265483 for n >= 0, record the primes.

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A096372 List of primes produced by a certain "prime-generating" quartic polynomial.

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Maple

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A076808 a(n) = 82n^3 - 1228n^2 + 6130n - 5861.

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