A074528 -id:A074528 - cp's OEIS frontend

A125706 Numbers k such that 6^k + 3^k + 2^k = A074528(k) is a prime.

0, 1, 3, 9, 13, 25, 27, 29, 95, 107, 153, 159, 675, 1633, 1693, 2337, 25998, 64289

Offset: 1

Alexander Adamchuk, Jan 31 2007

Corresponding primes of the form 6^k + 3^k + 2^k are {3, 11, 251, 10097891, 13062296531, 28430288877251865251, 1023490376703200952251, 36845653355419807219091, 84017312692910353150294530243640676594723260464784043666542128557055486251, ...}.

Cf. A074528.

Do[f=6^n+3^n+2^n;If[PrimeQ[f],Print[{n,f}]],{n,1,675}]

More terms from Ryan Propper, Mar 30 2007

a(18) from Michael S. Branicky, Jul 28 2024

A291556 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = (n!)^k * Sum_{i=1..n} 1/i^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 5, 11, 4, 0, 1, 9, 49, 50, 5, 0, 1, 17, 251, 820, 274, 6, 0, 1, 33, 1393, 16280, 21076, 1764, 7, 0, 1, 65, 8051, 357904, 2048824, 773136, 13068, 8, 0, 1, 129, 47449, 8252000, 224021776, 444273984, 38402064, 109584, 9

Offset: 0

Seiichi Manyama, Aug 26 2017

			Square array begins:
   0,  0,   0,     0,      0, ...
   1,  1,   1,     1,      1, ...
   2,  3,   5,     9,     17, ...
   3, 11,  49,   251,   1393, ...
   4, 50, 820, 16280, 357904, ...

Seiichi Manyama, Antidiagonals n = 0..59, flattened

Columns k=0-10 give: A001477, A000254, A001819, A066989, A203229, A099827, A291456, A291505, A291506, A291507, A291508.
Rows n=0-3 give: A000004, A000012, A000051, A074528.
Main diagonal gives A060943.

A:= (n, k)-> n!^k * add(1/i^k, i=1..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 26 2017

A[0, ] = 0; A[1, ] = 1; A[n_, k_] := A[n, k] = ((n-1)^k + n^k) A[n-1, k] - (n-1)^(2k) A[n-2, k];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 11 2019 *)

A(0, k) = 0, A(1, k) = 1, A(n+1, k) = (n^k+(n+1)^k)*A(n, k) - n^(2*k)*A(n-1, k).

A160449 Array read by antidiagonals: T(n,k) is the number of isomorphism classes of n-fold coverings of a connected graph with Betti number k (1 <= n, 0 <= k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 11, 8, 1, 1, 7, 43, 49, 16, 1, 1, 11, 161, 681, 251, 32, 1, 1, 15, 901, 14721, 14491, 1393, 64, 1, 1, 22, 5579, 524137, 1730861, 336465, 8051, 128, 1, 1, 30, 43206, 25471105, 373486525, 207388305, 7997683, 47449, 256, 1

Offset: 0

N. J. A. Sloane, Nov 13 2009

T(n,k) is the number of orbits of the conjugacy action of Sym(n) on Sym(n)^k [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 25 2015

			The array begins:
      k=0   k=1   k=2   k=3     k=4       k=5
  n=1   1     1     1     1       1         1
  n=2   1     2     4     8      16        32
  n=3   1     3    11    49     251      1393
  n=4   1     5    43   681   14491    336465
  n=5   1     7   161 14721 1730861 207388305

Álvar Ibeas, Table of n, a(n) for n = 0..1829
Michael W. Hero and Jeb F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Mathematics, 309 (2009), 6508-6514. See Table 3.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See Table 2.

Rows: A000012, A000079, A074528, A160450, A160454, A176709.
Columns: A000041, A110143, A152612, A160446, A160447, A160448.
Cf. A057004.

def A160449(n, k):
    return sum(p.aut()**(k - 1) for p in Partitions(n))
# Álvar Ibeas, Mar 25 2015

Name clarified and more terms added by Álvar Ibeas, Mar 25 2015

A246985 Expansion of (1-8x+14x^2)/((1-2x)(1-3x)(1-6*x)).

Original entry on oeis.org

1, 3, 11, 49, 251, 1393, 8051, 47449, 282251, 1686433, 10097891, 60526249, 362976251, 2177317873, 13062296531, 78368963449, 470199366251, 2821153019713, 16926788715971, 101560344351049, 609360902796251, 3656161927895953, 21936961102828211, 131621735227521049, 789730317205170251, 4738381620767930593

Offset: 0

N. J. A. Sloane, Sep 15 2014

From Álvar Ibeas, Mar 25 2015: (Start)
Number of isomorphism classes of 3-fold coverings of a connected graph with circuit rank n [Kwak and Lee].
Number of orbits of the conjugacy action of Sym(3) on Sym(3)^n [Kwak and Lee].
(End)

J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
A. Prasad, Equivalence classes of nodes in trees and rational generating functions, arXiv:1407.5284 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Apart from first term, same as A074528. Third row of A160449.

[n le 3 select 2*Factorial(n)-1 else 11*Self(n-1)-36*Self(n-2)+36*Self(n-3): n in [1..30]];

Table[2^(n - 1) + 3^(n - 1) + 6^(n - 1), {n, 0, 30}] (* Bruno Berselli, Mar 25 2015 *)
LinearRecurrence[{11,-36,36},{1,3,11},30] (* Harvey P. Dale, Jan 17 2019 *)

Vec((1-8*x+14*x^2)/((1-2*x)*(1-3*x)*(1-6*x)) + O(x^30)) \\ Michel Marcus, Jan 14 2016

G.f.: (1-8*x+14*x^2)/((1-2*x)*(1-3*x)*(1-6*x)).
a(n) = 11*a(n-1) - 36*a(n-2) + 36*a(n-3) for n>2. [Bruno Berselli, Mar 25 2015]
a(n) = 2^(n-1) + 3^(n-1) + 6^(n-1). - Álvar Ibeas, Mar 25 2015

Signature corrected and Ibeas formula adapted to the offset by Bruno Berselli, Mar 25 2015

cp's OEIS Frontend

A125706 Numbers k such that 6^k + 3^k + 2^k = A074528(k) is a prime.

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A291556 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = (n!)^k * Sum_{i=1..n} 1/i^k.

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A160449 Array read by antidiagonals: T(n,k) is the number of isomorphism classes of n-fold coverings of a connected graph with Betti number k (1 <= n, 0 <= k).

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A246985 Expansion of (1-8*x+14*x^2)/((1-2*x)*(1-3*x)*(1-6*x)).

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A246985 Expansion of (1-8x+14x^2)/((1-2x)(1-3x)(1-6*x)).