A053598 -id:A053598 - cp's OEIS frontend

A054576 Largest proper factor of the largest proper factor of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 5, 1, 1, 1, 6, 1, 1, 3, 7, 1, 5, 1, 8, 1, 1, 1, 9, 1, 1, 1, 10, 1, 7, 1, 11, 5, 1, 1, 12, 1, 5, 1, 13, 1, 9, 1, 14, 1, 1, 1, 15, 1, 1, 7, 16, 1, 11, 1, 17, 1, 7, 1, 18, 1, 1, 5, 19, 1, 13, 1, 20, 9, 1, 1, 21, 1, 1, 1, 22, 1, 15, 1, 23, 1, 1, 1, 24

Offset: 1

Henry Bottomley, Apr 11 2000

Here a "proper factor of n" means 1 if n = 1, and otherwise any d that divides n with 1 <= d < n. - N. J. A. Sloane, Dec 26 2022

			The largest proper factor of 8 is 4, the largest proper factor of 4 is 2, so a(8) = 2. - _N. J. A. Sloane_, Dec 26 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A053598.

Cf. A020639, A032742, A033942, A037143, A117357, A117358.

a054576 = a032742 . a032742  -- Reinhard Zumkeller, Jan 31 2015

A054576(n)=A032742(A032742(n)) \\ M. F. Hasler, Apr 03 2017

a(n) = A053598(A053598(n))

a(n) = A032742(A032742(n)); A117357(n) = A020639(a(n)); A117358(n) = A032742(a(n)) = a(n) / A117357(n); a(A037143(n)) = 1, a(A033942(n)) > 1. - Reinhard Zumkeller, Mar 10 2006

Deleted an incorrect comment and link. - N. J. A. Sloane, Dec 26 2022

A053418 Number of unlabeled directed graphs with n arcs and no isolated vertices.

Original entry on oeis.org

1, 1, 5, 17, 80, 365, 1981, 11222, 69511, 455663, 3169244, 23170347, 177513359, 1418920570, 11798710013, 101778754655, 908722427531, 8380602471646, 79692654473866, 780142956502644, 7851084073063731, 81120767066417308

Offset: 0

Vladeta Jovovic, Jan 10 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

The labeled version is A121252.
Column sums of A350908.
Cf. A000273, A000664, A053454, A053598 (by # of nodes).

Euler transform of A053454. - Andrew Howroyd, Jan 28 2022

Edited and extended by Max Alekseyev, Sep 18 2009

A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 3, 4, 4, 1, 1, 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1, 0, 0, 0, 3, 34, 116, 331, 669, 1128, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 1, 15, 134, 664, 2535, 7796, 19719, 42193, 77324, 122960, 170317, 206983, 220768

Offset: 1

Andrew Howroyd, Jan 28 2022

			Triangle begins:
  [1] 0;
  [2] 0, 1, 1;
  [3] 0, 0, 3, 4,  4,  1,  1;
  [4] 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums are A053598.
Column sums are A053418.
The labeled version is A054547.
Cf. A052283, A054733.

\\ See A054733 for G.
row(n)={Vecrev(G(n,y)-G(n-1,y), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) }

A361588 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with k strongly connected components and without isolated nodes.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 5, 4, 4, 0, 83, 57, 37, 25, 0, 5048, 2411, 1110, 550, 271, 0, 1047008, 325015, 101467, 37140, 15024, 5682, 0, 705422362, 136887749, 27765860, 7139149, 2259378, 780314, 237684, 0, 1580348371788, 183852357683, 23088181536, 3923330808, 907186816, 258971872, 78716548, 20042357

Offset: 0

Andrew Howroyd, Mar 16 2023

			Triangle begins:
  1;
  0,       0;
  0,       1,      1;
  0,       5,      4,      4;
  0,      83,     57,     37,    25;
  0,    5048,   2411,   1110,   550,   271;
  0, 1047008, 325015, 101467, 37140, 15024, 5682;
  ...

Column k=1 is A035512.
Main diagonal is A361589.
Row sums are A053598.
Cf. A350794, A361582, A361587, A361590.

\\ See PARI link in A350794 for program code.
{ my(A=A361588triang(6)); for(n=1, #A, print(A[n])) }

T(n,k) = A361582(n,k) - A361582(n-1,k-1).

A217654 Triangular array read by rows. T(n,k) is the number of unlabeled directed graphs of n nodes that have exactly k isolated nodes.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 13, 2, 0, 1, 202, 13, 2, 0, 1, 9390, 202, 13, 2, 0, 1, 1531336, 9390, 202, 13, 2, 0, 1, 880492496, 1531336, 9390, 202, 13, 2, 0, 1, 1792477159408, 880492496, 1531336, 9390, 202, 13, 2, 0, 1, 13026163465206704, 1792477159408, 880492496, 1531336, 9390, 202, 13, 2, 0, 1

Offset: 0

Geoffrey Critzer, Oct 09 2012

Row sums give A000273.

Column k = 0 is A053598.

			Triangle T(n,k) begins:
        1;
        0,    1;
        2,    0,   1;
       13,    2,   0,  1;
      202,   13,   2,  0, 1;
     9390,  202,  13,  2, 0, 1;
  1531336, 9390, 202, 13, 2, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..43, flattened

Cf. A000273, A053598.

b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(p[j]-1+add(
      igcd(p[k], p[j]), k=1..j-1)*2, j=1..nops(p)))([l[], 1$n])),
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
    end:
g:= proc(n) option remember; b(n$2, []) end:
T:= (n, k)-> g(n-k)-`if`(kAlois P. Heinz, Sep 04 2019

Needs["Combinatorica`"]; f[list_]:=Insert[Select[list,#>0&],0,-2]; nn=10; s=Sum[NumberOfDirectedGraphs[n]x^n, {n,0,nn}]; Drop[Flatten[Map[f, CoefficientList[Series[s (1-x)/(1-y x), {x,0,nn}], {x,y}]]], 1]
(* Second program: *)
b[n_, i_, l_List] := If[n==0 || i==1, 1/n!*2^(Function[p, Sum[p[[j]] - 1 + Sum[GCD[p[[k]], p[[j]]], {k, 1, j - 1}]*2, {j, 1, Length[p]}]][Join[l, Array[1&, n]]]), Sum[b[n - i*j, i - 1, Join[l, Array[i&, j]]]/j!/i^j, {j, 0, n/i}]];
g[n_] := g[n] = b[n, n, {}];
T[n_, k_] := g[n - k] - If[k < n, g[n - k - 1], 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

O.g.f.: A(x)*(1-x)/(1-y*x) where A(x) is o.g.f. for A000273.

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A054576 Largest proper factor of the largest proper factor of n.

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A053418 Number of unlabeled directed graphs with n arcs and no isolated vertices.

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A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

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A361588 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with k strongly connected components and without isolated nodes.

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A217654 Triangular array read by rows. T(n,k) is the number of unlabeled directed graphs of n nodes that have exactly k isolated nodes.

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