cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A060179 Sum of distinct orders of degree-n permutations.

Original entry on oeis.org

1, 1, 3, 6, 10, 21, 21, 50, 73, 116, 167, 248, 385, 496, 728, 959, 1548, 1899, 2835, 3609, 5042, 6403, 8336, 12187, 15522, 21358, 26090, 35298, 44147, 62512, 76289, 101403, 123883, 156880, 200086, 254175, 335380, 413184, 505860, 615258, 810767, 980747, 1293953
Offset: 0

Views

Author

Vladeta Jovovic, Mar 19 2001

Keywords

Examples

			Set of orders of all degree 7 permutations is {1,2,3,4,5,6,7,10,12} so a(7)=1+2+3+4+5+6+7+10+12=50.

Links

Crossrefs

Cf. A060014, A060015.
Cf. A009490.
Row sums of A256553.

Programs

  • Maple
    b:= proc(n, i) option remember; (p->`if`(i*n=0, 1,
       add(b(n-p^j, i-1)*p^j, j=1..ilog[p](n))+
         b(n, i-1)))(`if`(i=0, 0, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 12 2017
  • Mathematica
    b[n_, i_] := b[n, i] = Function [p, If[i*n == 0, 1, Sum[b[n-p^j, i-1]*p^j, {j, 1, Floor@Log[p, n]}] + b[n, i-1]]][If[i == 0, 0, Prime[i]]];
a[n_] := b[n, PrimePi[n]];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)

Formula

G.f.: Prod(p prime, 1 + Sum(k >= 1, p^k*x^(p^k))) / (1-x). - Vladeta Jovovic, Sep 18 2002

Extensions

More terms from David Wasserman, May 29 2002
a(0)=1 prepended by Alois P. Heinz, Apr 01 2015

A020902 Number of nonisomorphic cyclic subgroups of alternating group A_n (or number of distinct orders of even permutations of n objects); number of different LCM's of partitions of n which have even number of even parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 18, 22, 26, 30, 35, 39, 46, 51, 60, 67, 76, 84, 94, 105, 119, 133, 147, 162, 176, 196, 218, 240, 263, 286, 310, 340, 374, 409, 441, 476, 515, 559, 608, 662, 711, 762, 817, 883, 955, 1030, 1104, 1177, 1257, 1352, 1453, 1559
Offset: 0

Views

Author

Vladeta Jovovic

Keywords

Examples

			a(8)=8 because lcm{1^8} = 1, lcm{1^4 * 2^2, 2^4} = 2, lcm{1^5 * 3^1, 1^2 * 3^2} = 3, lcm{4^2, 1^2 * 2^1 * 4^1} = 4, lcm{1^3 * 5^1} = 5, lcm{2^1 * 6^1, 1^1 * 2^2 * 3^1} = 6, lcm{1^1 * 7^1} = 7, lcm{3^1 * 5^1} = 15.

References

  • V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Crossrefs

Cf. A034891.

Formula

a(n) = A009490(n-2) + A035942(n-1) + A035942(n), n > 1, a(0)=a(1)=1.

A211392 The number of divisors d of n! such that the symmetric group on n letters contains no elements of order d.

Original entry on oeis.org

0, 0, 1, 4, 10, 24, 51, 85, 146, 254, 520, 769, 1557, 2561, 3997, 5333, 10705, 14633, 29315, 40970, 60722, 95912, 191902, 242769, 339909, 532088, 677224, 917112, 1834373, 2332596, 4665375, 5529352, 7864049, 12164824, 16422587, 19595164, 39190653, 60465758
Offset: 1

Views

Author

Alexander Gruber, Feb 07 2013

Keywords

Links

Crossrefs

Cf. A000005, A009490, A027423, A211391.

Programs

  • Maple
    b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, b(n, i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n)))
    end:
a:= n-> numtheory[tau](n!) -b(n, numtheory[pi](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, b[n, i-1] + Sum[b[n-p^j, i-1], {j, 1, Floor@Log[p, n]}]]];
a[n_] := DivisorSigma[0, n!] - b[n, PrimePi[n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

Formula

a(n) = A000005(n!) - A009490(n).

Extensions

More terms from Alois P. Heinz, Feb 11 2013

A225725 Triangle of transformation semigroup sizes generated by a single element.

Original entry on oeis.org

1, 1, 3, 1, 10, 15, 2, 41, 129, 80, 6, 196, 1115, 1260, 510, 24, 20, 1057, 10395, 17780, 12840, 3744, 840, 6322, 105315, 258510, 264810, 135492, 47250, 4920, 0, 0, 504, 0, 420, 41393, 1160635, 4018000, 5318180, 3788400, 1837024, 513120, 38640, 0, 32256, 0, 26880, 0, 0, 2688
Offset: 0

Views

Author

Chad Brewbaker, May 14 2013

Keywords

Comments

If you take the powers of a finite function you generate a lollipop graph. A222029 organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is this triangle.

Examples

			T(1,1) = #{[0]} = 1.
T(2,1) = #{[0,1], [0,0], [1,1]} = 3.
T(2,2) = #{[1,0]} = 1.
Triangle begins:
:    1;
:    1;
:    3,      1;
:   10,     15,      2;
:   41,    129,     80,      6;
:  196,   1115,   1260,    510,     24,    20;
: 1057,  10395,  17780,  12840,   3744,   840;
: 6322, 105315, 258510, 264810, 135492, 47250, 4920, 0, 0, 504, 0, 420;

Links

Crossrefs

First column is A000248.
Row sums are: A000312.
Row lengths are A000793.
Number of nonzero elements of rows give A009490.
Cf. A222029.

Programs

  • Ruby
    # See Brewbaker link.

Extensions

More terms, some terms corrected by Alois P. Heinz, Aug 17 2017
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