Stephen A. Silver - cp's OEIS frontend

User: Stephen A. Silver

Stephen A. Silver has authored 3 sequences.

A214003 Number of degree-n permutations of prime order.

0, 1, 5, 17, 69, 299, 1805, 9099, 37331, 205559, 4853529, 49841615, 789513659, 9021065871, 70737031469, 420565124399, 22959075244095, 385032305178719, 10010973102879761, 152163983393187399, 1498273284120348539, 15639918041915598815, 1296204202723400597109

Offset: 1

Stephen A. Silver, Feb 15 2013

nonn

			The symmetric group S_5 has 25 elements of order 2, 20 elements of order 3, and 24 elements of order 5. All other elements are of nonprime order (1, 4, or 6), so a(5) = 25 + 20 + 24 = 69.

Stephen A. Silver, Table of n, a(n) for n = 1..451

Cf. A001189, A001471, A057731, A059593, A153760, A153761, A186202.

b:= proc(n,p) option remember;
      `if`(n add(b(n, ithprime(i)), i=1..numtheory[pi](n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 16 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, ilcm(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Jan 19 2023

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n],PrimeQ[Apply[LCM, #]] &]]], {n, 1,23}] (* Geoffrey Critzer, Nov 08 2015 *)

a(n) = Sum_{p prime} A057731(n,p).

E.g.f.: exp(x)*Sum_{p in Primes} exp(x^p/p)-1. - Geoffrey Critzer, Nov 08 2015

A055397 Maximum population of an n X n stable pattern in Conway's Game of Life.

Original entry on oeis.org

0, 4, 6, 8, 16, 18, 28, 36, 43, 54, 64, 76, 90, 104, 119, 136, 152, 171, 190, 210, 232, 253, 276, 301, 326, 352, 379, 407, 437, 467, 497, 531, 563, 598, 633, 668, 706, 744, 782, 824, 864, 907, 949, 993, 1039, 1085, 1132, 1181, 1229, 1280, 1331, 1382, 1436

Offset: 1

Stephen A. Silver, Jun 25 2000

nonn

			a(3) = 6 because a ship has 6 cells and no other 3 X 3 stable pattern has more.

G. Chu and P. J. Stuckey, A complete solution to the Maximum Density Still Life Problem, Artificial Intelligence, 184:1-16 (2012).
G. Chu, K. E. Petrie, and N. Yorke-Smith, Constraint Programming to Solve Maximal Density Still Life, In Game of Life Cellular Automata chapter 10, A. Adamatzky, Springer-UK, 99-114 (2010).
G. Chu, P. Stuckey, and M.G. de la Banda, Using relaxations in Maximum Density Still Life, In Proc. of Fifteenth Intl. Conf. on Principles and Practice of Constraint Programming, 258-273 (2009).
Stephen Silver, Dense Stable Patterns

a(n) = (n^2)/2 + O(n).

For n >= 55, floor(n^2/2 + 17*n/27 - 2) <= a(n) <= ceiling(n^2/2 + 17*n/27 - 2), which gives all values of this sequence within +- 1.

a(11)-a(27) from Nathaniel Johnston, May 15 2011, based on table in Chu et al.

a(28)-a(53) from Nathaniel Johnston, Nov 27 2013, based on work by Chu et al.

A048648 Order of n-th stable homotopy group of spheres.

Original entry on oeis.org

2, 2, 24, 1, 1, 2, 240, 4, 8, 6, 504, 1, 3, 4, 960, 4, 16, 16, 528, 24, 4, 4, 3144960, 4, 4, 12, 24, 2, 3, 6, 65280, 16, 32, 32, 114912, 6, 12, 120, 1267200, 384, 32, 96, 552, 8, 5760, 48, 12579840, 64, 12, 24, 384, 24, 16, 8, 20880, 2, 8, 4, 687456, 4, 1

Offset: 1

Stephen A. Silver

Proved by Serre to be finite for all positive n.

The best current reference is Isaksen-Wang-Xu, Table 1. - Charles Rezk, Aug 22 2020

			Pi_1^S = Pi_4(S^3) = Z/2Z, so a(1) = |Z/2Z| = 2.

D. B. Fuks, "Spheres, homotopy groups of the", Encyclopaedia of Mathematics, Vol. 8.
S. O. Kochman, Stable homotopy groups of spheres. A computer-assisted approach. Lecture Notes in Mathematics, 1423. Springer-Verlag, Berlin, 1990. 330 pp. ISBN: 3-540-52468-1. [Math. Rev. 91j:55016]
Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, 2003.
Hirosi Toda, Composition Methods in Homotopy Groups of Spheres, Princeton University Press, 1962.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..83 (terms 1..81 from Charles Rezk, terms 82..83 using data from Isaksen, Wang & Xu (2023))
Kevin Hartnett, An Old Conjecture Falls, Making Spheres a Lot More Complicated, Quanta Magazine (2023)
A. Hatcher, Stable Homotopy Groups of Spheres
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publications mathématiques de l'IHÉS, 137 (2023), 107-243; arXiv:2001.04511 [math.AT], 2020-2023.
S. O. Kochman and M. E. Mahowald, On the computation of stable stems, The Cech Centennial (Boston, MA, 1993), 299-316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018]
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
Robert Scharein's program sphere-link.c linked from the Linked Spheres page [has incorrect a(23) and a(29)-a(33)]
Wikipedia, Homotopy groups of spheres

Cf. A001676.

a(n) = |Pi_n^S| = |Pi_{k+n}(S^k)| for k > n+1.

More terms from Alex Fink (finka(AT)math.ucalgary.ca), Aug 10 2006
a(23) and a(29)-a(33) corrected by Charles Rezk, Aug 22 2020
More terms from Charles Rezk, Aug 25 2020

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User: Stephen A. Silver

A214003 Number of degree-n permutations of prime order.

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A055397 Maximum population of an n X n stable pattern in Conway's Game of Life.

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A048648 Order of n-th stable homotopy group of spheres.

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