A055869 -id:A055869 - cp's OEIS frontend

A098463 Numbers k such that A055869(k) = (k+1)^k - k^k is prime.

2, 3, 5, 7, 167

Offset: 1

Hugo Pfoertner, Sep 14 2004

The next term is > 5000.

			a(2) = 3 because (3+1)^3 - 3^3 = 4^3 - 3^3 = 64 - 27 = 37 is prime.

Cf. A055869 ((n+1)^n-n^n), A085682 (k^k-(k-1)^k is prime).

[n: n in [0..190]| IsPrime((n+1)^n -n^n)]; // Vincenzo Librandi, Jun 22 2014

Select[Range[300], PrimeQ[(#+1)^#-#^#]&] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

isok(k) = ispseudoprime((k+1)^k - k^k); \\ Jinyuan Wang, Mar 19 2020

A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

Original entry on oeis.org

0, 1, 4, 21, 156, 1555, 19608, 299593, 5380840, 111111111, 2593742460, 67546215517, 1941507093540, 61054982558011, 2085209001813616, 76861433640456465, 3041324492229179280, 128583032925805678351, 5784852794328402307380, 275941052631578947368421

Offset: 1

Henry Bottomley, Feb 21 2001

nonn

(n-1)-digit repunits in base n written in decimal.

			a(10)=111111111; i.e., just nine 1's (converted from base 10 to decimal).

Harry J. Smith, Table of n, a(n) for n=1..200

Cf. A000169, A037205, A055869, A060073, A125118, A228275.

Cf. other sequences of generalized repunits, such as A053696, A055129, A031973, A125598, A173468, A023037, A119598, A085104, and A162861.

[0] cat [ (n^(n-1) -1)/(n-1) : n in [2..25]]; // G. C. Greubel, Aug 15 2022

Join[{0},Array[(#^(#-1)-1)/(#-1)&,20,2]] (* Harvey P. Dale, Jun 04 2013 *)

a(n) = if (n==1, 0, (n^(n - 1) - 1)/(n - 1)); \\ Harry J. Smith, Jul 01 2009

[0]+[(n^(n-1) -1)/(n-1) for n in (2..25)] # G. C. Greubel, Aug 15 2022

a(n+1) = Sum_{k=1..n} n^(k-1)*C(n, k). - Olivier Gérard, Jun 26 2001 [Corrected by Mathew Englander, Dec 15 2020]
a(n) = Sum_{j=2..n} n^(n-j). - Zerinvary Lajos, Sep 11 2006
a(n+1) = A125118(n,n). - Reinhard Zumkeller, Nov 21 2006
a(n) = Integral_{x=1/n..1} 1/x^n dx. - Francesco Daddi, Aug 01 2011
a(n) = A037205(n-1)/(n-1) = A060073(n)*(n-1) = A023037(n) - A000169(n).
a(n) = [x^n] x^2/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017
a(n) = 1 + A228275(n, n-2) for n >= 2. - Mathew Englander, Dec 14 2020

Name edited by Michel Marcus, Dec 14 2020

A060226 a(n) = n^n - n*(n-1)^(n-1).

Original entry on oeis.org

1, 0, 2, 15, 148, 1845, 27906, 496951, 10188872, 236425545, 6125795110, 175311670611, 5492360400924, 186965800764925, 6871755333266474, 271213787997489135, 11440441827615801616, 513645612633274386705

Offset: 0

Henry Bottomley, Jul 12 2001

nonn

For n > 0, a(n) = number of endofunctions of [n] mapping some x<>1 to 1. - Len Smiley, Nov 15 2001 (Endofunction interpretation from a(n) = n*(n^(n-1) - (n-1)^(n-1)).)

Harry J. Smith, Table of n, a(n) for n = 0..100
D. Callan, A Bijection between Marked Trees
Leonard Smiley, Problem 10781, Amer. Math. Monthly, 107, Feb. 2000, p. 176.

Cf. A000312, A045531, A055869.

a060226 0 = 1
a060226 n = a000312 n - n * a000312 (n - 1)
-- Reinhard Zumkeller, Aug 27 2012

A060226:= func< n | n^n - n*(n-1)^(n-1) >;
[A060226(n): n in [0..30]]; // G. C. Greubel, Nov 03 2024

f := n-> n*sum(binomial(n-1,j-1)*(n-1)^(n-j), j=2..n);
g := n-> n^n -n*(n-1)^(n-1);
h := n-> sum(binomial(n,j)*j^(j-1)*(n-j)^(n-j), j=2..n);
k := n-> sum(binomial(n,j-1)*(j-1)^(j-1)*(n-j)^(n-j), j=2..n); # then a(n)=f(n)=g(n)=h(n)=k(n)

Join[{1,0},Table[n^n-n*(n-1)^(n-1),{n,2,20}]] (* Harvey P. Dale, Nov 16 2012 *)

{ for (n=0, 100, write("b060226.txt", n, " ", n^n - n*(n - 1)^(n - 1)); ) } \\ Harry J. Smith, Jul 03 2009

def A060226(n): return n^n - n*(n-1)^(n-1)
[A060226(n) for n in range(31)] # G. C. Greubel, Nov 03 2024

a(n) = n*A055869(n-1).
Limit_{n -> oo} ( a(n)/a(n-1) - a(n-1)/a(n-2) ) -> e.
E.g.f.: (1-x)/(1-T), where T=T(x) is Euler's tree function (see A000169). The e.g.f. for n > 0 terms only (applicable to endofunctions) is (T - x)/(1 - T). - Len Smiley, Dec 10 2001

A085529 a(n) = (2n+1)^(2n+1).

Original entry on oeis.org

1, 27, 3125, 823543, 387420489, 285311670611, 302875106592253, 437893890380859375, 827240261886336764177, 1978419655660313589123979, 5842587018385982521381124421, 20880467999847912034355032910567, 88817841970012523233890533447265625, 443426488243037769948249630619149892803

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

a(n) == 2*n + 1 (mod 24). - Mathew Englander, Aug 16 2020

Cf. A000312, A005408, A016754, A085527, A085528, A085530, A085531, A085532, A085533, A085534, A085535.

Cf. A073009, A083648.

#^#&/@Range[1,31,2] (* Harvey P. Dale, Oct 05 2019 *)

From Mathew Englander, Aug 16 2020: (Start)
a(n) = A000312(2*n + 1).
a(n) = A016754(n)^n * (2*n + 1).
a(n) = A085527(n)^2 * (2*n + 1).
a(n) = A085528(n)^2 / (2*n + 1).
a(n) = A085530(n) * A005408(n).
a(n) = A085531(n) * A016754(n).
a(n) = A085532(n)^2 - A215265(2*n + 1).
a(n) = A085533(n) + A045531(2*n + 1).
a(n) = A085534(n+1) - A007781(2*n + 1).
a(n) = A085535(n+1) - A055869(2*n + 1).
(End)
Sum_{n>=0} 1/a(n) = (A073009 + A083648)/2 = 1.0373582538... . - Amiram Eldar, May 17 2022

A086877 Primes of the form (k+1)^k - k^k.

Original entry on oeis.org

5, 37, 4651, 1273609

Offset: 1

Cino Hilliard, Aug 21 2003

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..5

The values of k are in A098463.
Primes in A055869.
Cf. A085682.

f(n) = for(x=1,n,y=(x+1)^x-x^x; if(isprime(y),print1(y, ", ")))

a(n) = A055869(A098463(n)). - Elmo R. Oliveira, Feb 19 2025

A085283 a(n) = nn^n - (n-1)(n-1)^n.

Original entry on oeis.org

1, 1, 7, 65, 781, 11529, 201811, 4085185, 93864121, 2413042577, 68618940391, 2138428376721, 72470493235141, 2653457921150425, 104382202543721467, 4390455017903519489, 196621779843659466481, 9340717969198079777313

Offset: 0

Paul Barry, Jun 26 2003

The system of equations
x(0) = n*x(1) + 1,
(n-1)*x(1) = n*x(2) + 1,
...
(n-1)*x(n) = n*x(n+1) + 1.
relates to the Monkey-And-Coconuts problem and reduces to the single equation
A007778(n-1)*x(0) = A007778(n)*x(n+1) + a(n),
whose solutions {x(0),x(n+1)} are given by {A014293(n), A085606(n)=A007778(n-1) - 1}. - Lekraj Beedassy, Jul 15 2003
For n >= 1, a(n) is equal to the number of functions f: {1,2,...,n+1}->{1,2,...,n} such that Im(f) contains a fixed element. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
BBC, The Monkey and the Coconuts
K. Belcourt, How Many Coconuts
Santo D'Agostino, "The Coconut Problem"; Updated With Solution, May 2011.
J. Dean, Sailors, monkey and coconuts
A. K. Dewdney, The Monkey and the Coconuts
G. Lewandrowski, The Monkey Problem
R. Raja, Monkeys and Coconuts
A. Rishi, Nemo's sailors and the monkey
Dr. Rob, The MathForum, Coconut piles
D. J. Wright, Five Pirates and a Monkey

Cf. A045531, A055869.

Join[{1},Table[n*n^n-(n-1)(n-1)^n,{n,20}]] (* Harvey P. Dale, Sep 08 2016 *)

E.g.f.: -(x + 2*x*W(-x) + W(-x)^2)/(W(-x)*(1 + W(-x))^3), where W(x) is the Lambert W function. - Fabian Pereyra, Sep 26 2023

A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 9, 5, 1, 0, 64, 37, 9, 1, 0, 625, 369, 97, 14, 1, 0, 7776, 4651, 1275, 205, 20, 1, 0, 117649, 70993, 19981, 3410, 380, 27, 1, 0, 2097152, 1273609, 365001, 64701, 7770, 644, 35, 1, 0, 43046721, 26269505, 7628545, 1388310, 174951, 15834, 1022, 44, 1

Offset: 0

Werner Schulte, Feb 15 2023

Triangle T is created using 2-Stirling numbers of the first (A049444) and the second (A143494) kind. The unusual construction is as follows:
  Define A(n, k) by recurrence A(n, k) = A(n-1, k-1) + (k+1) * A(n-1, k) for 0 < k < n with initial values A(n, n) = 1, n >= 0, and A(n, 0) = 0, n > 0. A without column k = 0 is A143494. Let B = A^(-1) matrix inverse of A. B without column k = 0 is A049444. Now define T(m, k) = Sum_{i=0..m-k} B(m-k, i) * A(m-1+i, m-1) for 0 < k <= m = n/2 and T(m, 0) = 0^m for 0 <= m = n/2; T(i, j) = 0 if i < j or j < 0.
Matrix inverse of T is A360753. - Werner Schulte, Feb 21 2023
Conjecture: the transpose of this array is the upper triangular matrix U in the LU factorization of the array of Stirling numbers of the second kind read as a square array; the corresponding lower triangular array L is the triangle of Stirling numbers of the second kind. See the example section below. - Peter Bala, Oct 10 2023

			Triangle T(n, k), 0 <= k <= n, starts:
n\k :  0         1         2        3        4       5      6     7   8  9
==========================================================================
  0 :  1
  1 :  0         1
  2 :  0         2         1
  3 :  0         9         5        1
  4 :  0        64        37        9        1
  5 :  0       625       369       97       14       1
  6 :  0      7776      4651     1275      205      20      1
  7 :  0    117649     70993    19981     3410     380     27     1
  8 :  0   2097152   1273609   365001    64701    7770    644    35   1
  9 :  0  43046721  26269505  7628545  1388310  174951  15834  1022  44  1
  etc.
From _Peter Bala_, Oct 10 2023: (Start)
LU factorization of the square array of Stirling numbers of the second kind (apply Xu, Lemma 2.2):
 / 1               \ / 1   1   1   1  ...\    / 1   1   1    1  ... \
 | 1   1           ||      2   5   9  ...|   |  1   3   6   10  ... |
 | 1   3   1       ||          9  37  ...| = |  1   7  25   65  ... |
 | 1   7   6   1   ||             64  ...|   |  1  15  90  350  ... |
 | ...             ||                 ...|   |  ...                 |
(End)

Wikipedia, LU decomposition
Aimin Xu, Determinants Involving the Numbers of the Stirling-Type, Filomat 33:6 (2019), 1659-1666.

Cf. A049444, A143494, A354794, A354795, A088956, A094587.
Cf. A000007 (column 0), A000169 (column 1), A055869 (column 2).
Cf. A000012 (main diagonal), A000096 (1st subdiagonal), A360753 (matrix inverse).

tabl(m) = {my(n=2*m, A = matid(n), B, T); for( i = 2, n, for( j = 2, i, A[i, j] = A[i-1, j-1] + j * A[i-1, j] ) ); B = A^(-1); T = matrix( m, m, i, j, if( j == 1, 0^(i-1), sum( r = 0, i-j, B[i-j+1, r+1] * A[i-1+r, i-1] ) ) ); }

For the definition of triangle T see Comments section.
Conjectured formulas:
T(n, k) = (Sum_{i=k..n} A354794(n, i) * (i-1)!) / (k-1)! for 0 < k <= n.
T(n, k) - k * T(n, k+1) = A354794(n, k) for 0 <= k <= n.
T(n, 1) = A000169(n) = n^(n-1) for n > 0.
T(n, 2) = A055869(n-1) = n^(n-1) - (n-1)^(n-1) for n > 1.
T(n, k) = (Sum_{i=0..k-1} (-1)^i * binomial(k-1, i) * (n-i)^(n-1)) / (k-1)! for 0 < k <= n.
Sum_{i=1..n} (-1)^(n-i) * binomial(n-1+k, i-1) * T(n, i) * (i-1)! = (k-1)^(n-1) for n > 0 and k >= 0.
Matrix product of A354795 and T without column 0 equals A094587.
Matrix product of T and A354795 without column 0 equals A088956.
E.g.f. of column k > 0: Sum_{n>=k} T(n, k) * t^(n-1) / (n-1)! = (W(-t)/(-t)) * (Sum_{n>=k} A354794(n, k) * t^(n-1) / (n-1)!) where W is the Lambert_W-function.

A098462 a(n) = n^n + (n+1)^n.

Original entry on oeis.org

2, 3, 13, 91, 881, 10901, 164305, 2920695, 59823937, 1387420489, 35937424601, 1028320041299, 32214185570737, 1096589879846397, 40304932850948641, 1590815394987706351, 67107935949376420097, 3013151821625033296145, 143473758373207779108265

Offset: 0

Hugo Pfoertner, Sep 08 2004

nonn

Cf. A000169, A000312, A045531 (n^n-(n-1)^n), A055869 ((n+1)^n-n^n).

a(n) = n^n + (n+1)^n; \\ Jinyuan Wang, Mar 19 2020

From Alois P. Heinz, Mar 19 2020: (Start)
a(n) = A000312(n) + A000169(n+1).
E.g.f.: (x-LambertW(-x))/((1+LambertW(-x))*x). (End)

a(0)=2 prepended by Alois P. Heinz, Mar 19 2020

A174552 Triangular array T(n,k): The differences in the columns of A174551.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 8, 7, 6, 6, 81, 65, 50, 36, 24, 1024, 781, 570, 390, 240, 120, 15625, 11529, 8162, 5460, 3360, 1800, 720, 279936, 201811, 140070, 92526, 57120, 31920, 15120, 5040

Offset: 0

Geoffrey Critzer, Mar 22 2010

T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the image of f contains {1,2,...,k} but not k+1.
Row sums = n^n.
Columns are asymptotic to n^n (1/e)(1-(1/e))^k.
Sum k*T(n,k) appears to be A055869.

			Triangle begins:
1;
0, 1;
1, 1, 2;
8, 7, 6, 6;
81,65,50,36,24;
1024,781,570,390,240,120;
15625, 11529, 8162, 5460, 3360, 1800, 720...

Table[Append[(-1) Differences[ Table[Sum[(-1)^i Binomial[k, i] (n - i)^n, {i, 0, k}], {k, 0, n}]], n! ], {n, 0, 7}] // Grid

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A098463 Numbers k such that A055869(k) = (k+1)^k - k^k is prime.

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A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

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A060226 a(n) = n^n - n*(n-1)^(n-1).

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A085529 a(n) = (2n+1)^(2n+1).

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A086877 Primes of the form (k+1)^k - k^k.

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A085283 a(n) = n*n^n - (n-1)*(n-1)^n.

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A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).

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A085283 a(n) = nn^n - (n-1)(n-1)^n.