A121684 -id:A121684 - cp's OEIS frontend

A078779 Union of S, 2S and 4S, where S = odd squarefree numbers (A056911).

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 101

Offset: 1

Benoit Cloitre, Jan 11 2003

nonn

Numbers n such that the cyclic group Z_n is a DCI-group.
Numbers n such that A008475(n) = A001414(n).
A193551(a(n)) = A000026(a(n)) = a(n). - Reinhard Zumkeller, Aug 27 2011
Union of squarefree numbers and twice the squarefree numbers (A005117). - Reinhard Zumkeller, Feb 11 2012
The complement is A046790. - Omar E. Pol, Jun 11 2016

T. D. Noe, Table of n, a(n) for n = 1..7098
B. Alspach and M. Mishna, Enumeration of Cayley graphs and digraphs, Discr. Math., 256 (2002), 527-539.
M. Mishna, Home Page
M. Muzychuk, On Adam's conjecture for circulant graphs, Discr. Math. 167 (1997), 497-510.

Cf. A121176, A121684, A008475, A001414, A046790.

a078779 n = a078779_list !! (n-1)
a078779_list = m a005117_list $ map (* 2) a005117_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y     = x : m xs ys'
                           | x == y    = x : m xs ys
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Feb 11 2012, Aug 27 2011

is(n)=issquarefree(n/gcd(n,2)) \\ Charles R Greathouse IV, Nov 05 2017

a(n) = (Pi^2/7)*n + O(sqrt(n)). - Vladimir Shevelev, Jun 08 2016

Edited by N. J. A. Sloane, Sep 13 2006

A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

Original entry on oeis.org

1, 4, 2, 15, 9, 3, 64, 40, 16, 4, 325, 205, 85, 25, 5, 1956, 1236, 516, 156, 36, 6, 13699, 8659, 3619, 1099, 259, 49, 7, 109600, 69280, 28960, 8800, 2080, 400, 64, 8, 986409, 623529, 260649, 79209, 18729, 3609, 585, 81, 9, 9864100, 6235300, 2606500, 792100, 187300, 36100, 5860, 820, 100, 10

Offset: 1

Thomas Wieder, Aug 15 2006

The first column is A007526 = "the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}." E.g. for n=3 there are 15 subsets: {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}. These are subsets with a number of elements l=1,...,n. The second column excludes all subsets with l=n elements. For n=3 one has therefore only the 9 subsets {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}. The third column excludes all subsets with l>=n-1 elements. For n=3 one has therefore only the 3 subsets {a}, {b},{c}. See also A121684. The second column is A038156 = n!*Sum(1/k!, k=1..n-1). The first lower diagonal are the squares A000290 = n^2. The second lower diagonal (15, 40, 85...) is A053698 = n^3 + n^2 + n + 1. The row sum is A030297 = a(n) = n*(n+a(n-1)).

T(i, j) is the total number of ordered sets of size 1 to i-j+1 that can be created from i distinct items. - Manfred Boergens, Jun 22 2022

			Triangle T(i,j) begins:
       1
       4     2
      15     9     3
      64    40    16     4
     325   205    85    25    5
    1956  1236   516   156   36   6
   13699  8659  3619  1099  259  49  7
   ...

Cf. A007526, A030297, A053698, A038156, A000290, A121682.

Mirror of triangle A285268.

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, T(i-1, j)*i+i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m, 1, -1}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j == 0, 0, Floor[j! E - 1]];
T[i_, j_] = b[i] - i! b[j - 1]/(j - 1)!;
Table[T[i, j], {i, 24}, {j, i}] // Flatten
(* Manfred Boergens, Jun 22 2022 *)

From Manfred Boergens, Jun 22 2022: (Start)
T(i, j) = Sum_{k=1..i-j+1} i!/(i-k)! = Sum_{k=j-1..i-1} i!/k!.
Sum-free formula: T(i, j) = b(i) - i!*b(j-1)/(j-1)! where b(0)=0, b(j)=floor(j!*e-1) for j>0.
(End)

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

A121176 Union of {8, 9, 18}, S, 2S and 4S, where S = squarefree numbers (A005117).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92

Offset: 1

N. J. A. Sloane, Sep 13 2006

nonn

The asymptotic density of this sequence is 15/(2*Pi^2) (A323669). - Amiram Eldar, May 10 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A121684, A323669.

Union[Join[{9, 18}, Select[Range[100], (e = IntegerExponent[#,2]) <= 3 && SquareFreeQ[#/2^e] &]]] (* Amiram Eldar, May 10 2022 *)

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A078779 Union of S, 2S and 4S, where S = odd squarefree numbers (A056911).

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A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

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A121176 Union of {8, 9, 18}, S, 2S and 4S, where S = squarefree numbers (A005117).

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