A055651 -id:A055651 - cp's OEIS frontend

A007925 a(n) = n^(n+1) - (n+1)^n.

-1, -1, -1, 17, 399, 7849, 162287, 3667649, 91171007, 2486784401, 74062575399, 2395420006033, 83695120256591, 3143661612445145, 126375169532421599, 5415486851106043649, 246486713303685957375, 11877172892329028459041, 604107995057426434824791

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

From Mathew Englander, Jul 07 2020: (Start)
All a(n) are odd and for n even, a(n) == 3 (mod 4); for n odd and n != 1, a(n) == 1 (mod 4).
The correspondence between n and a(n) when considered mod 6 is as follows: for n == 0, 1, 2, or 3, a(n) == 5; for n == 4, a(n) == 3; for n == 5, a(n) == 1.
For all n, a(n)+1 is a multiple of n^2.
For n odd  and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs of the above, see the Englander link. (End)

			a(2) = 1^2 - 2^1 = -1,
a(4) = 3^4 - 4^3 = 17.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

T. D. Noe, Table of n, a(n) for n = 0..100
Mathew Englander, Notes on OEIS A007925
Sergio Silva, Teste Numerico, Item 3.
H. J. Smith, Contour Plot of z = x^y - y^x

Cf. A051442.

Cf. A166326, A099498, A141074, A174379, A123206, A045575, A082754.

A007925:=n->n^(n+1)-(n+1)^n: seq(A007925(n), n=0..25); # Wesley Ivan Hurt, Jan 10 2017

lst={};Do[AppendTo[lst, (n^(n+1)-((n+1)^n))], {n, 0, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
#^(#+1)-(#+1)^#&/@Range[0,20] (* Harvey P. Dale, Oct 22 2011 *)

A007925[n]:=n^(n+1)-(n+1)^n$ makelist(A007925[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n)=n^(n+1)-(n+1)^n \\ Charles R Greathouse IV, Feb 06 2017

Asymptotic expression for a(n) is a(n) ~ n^n * (n - e). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
From Mathew Englander, Jul 07 2020: (Start)
a(n) = A111454(n+4) - 1.
a(n) = A055651(n, n+1).
a(n) = A220417(n+1, n) for n >= 1.
a(n) = A007778(n) - A000169(n+1).
(End)
E.g.f.: LambertW(-x)/((1+LambertW(-x))*x)-LambertW(-x)/(1+LambertW(-x))^3. - Alois P. Heinz, Jul 04 2022

A123206 Primes of the form x^y - y^x, for x,y > 1.

Original entry on oeis.org

7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249

Offset: 1

Alexander Adamchuk, Oct 04 2006

nonn

These are the primes in A045575, numbers of the form x^y - y^x, for x,y > 1. This includes all primes from A122735, smallest prime of the form (n^k - k^n) for k>1.

If y=1 was allowed, any prime p could be obtained for x=p+1. This motivates to consider sequence A243100 of primes of the form x^(y+1)-y^x. - M. F. Hasler, Aug 19 2014

			The primes 6102977801 and 1490116119372884249 are of the form 5^y-y^5 (for y=14 and y=26) and therefore members of this sequence. The next larger primes of this form would have y > 4500 and would be much too large to be included. - _M. F. Hasler_, Aug 19 2014

T. D. Noe, Table of n, a(n) for n=1..101 (terms < 10^400)
H. Lifchitz & R. Lifchitz, PRP of the form x^y-y^x on primenumbers.net.

Cf. A045575, A122735, A078202, A082754, A055651, A094133.
A163319 is the subsequences for fixed x=3, A243114 for x=6.
Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.

N:= 10^100: # to get all terms <= N
A:= NULL:
for x from 2 while x^(x+1) - (x+1)^x <= N do
   for y from x+1 do
      z:= x^y - y^x;
      if z > N then break
      elif z > 0 and isprime(z) then A:=A, z;
      fi
od od:
{A}; # Robert Israel, Aug 29 2014

Take[Select[Intersection[Flatten[Table[Abs[x^y-y^x],{x,2,120},{y,2,120}]]],PrimeQ[ # ]&],25]
nn=10^50; n=1; t=Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num0&&PrimeQ[#]&]],nn]] (* Harvey P. Dale, Nov 23 2013 *)

a=[];for(S=1,199,for(x=2,S-2,ispseudoprime(p=x^(y=S-x)-y^x)&&a=concat(a,p)));Set(a) \\ May be incomplete in the upper range of values, i.e., beyond a given S=x+y, a larger S may yield a smaller prime (for small x). - M. F. Hasler, Aug 19 2014

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073

Offset: 0

Henry Bottomley, Jun 08 2000

Columns and rows are A000012 (apart from first term), A000027, A001580, A001585, A001589, A001593 etc. Diagonals include A013499 (apart from first two terms), A051442, A051489.
Cf. A055651.
Contribution from Franklin T. Adams-Watters, Oct 26 2009: (Start)
Rows/columns: A000027, A001580, A001585, A001589, A001593, A001594, A001596.
Main diagonal is 2 * A000312. More diagonals: A051442, A051489, A155539.
Cf. A076980, A156353, A156354. (End)

E.g.f. Sum(n,m, T(n,m)/(n! m!)) = e^(x e^y) + e^(y e^x). [From Franklin T. Adams-Watters, Oct 26 2009]

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625

Offset: 0

Henry Bottomley, Jul 02 2001

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1

Cf. A055651, A055652, A303990.

{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)

t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A220417 Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 3, 1, -1, -3, 4, 0, 0, 0, -4, 5, -7, -17, 17, 7, -5, 6, -28, -118, 0, 118, 28, -6, 7, -79, -513, -399, 399, 513, 79, -7, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, 9, -431, -6049, -13983, -7849, 7849, 13983, 6049, 431, -9, 10, -924, -18954, -61440, -61318, 0, 61318, 61440, 18954, 924, -10

Offset: 1

Boris Putievskiy, Dec 14 2012

			The table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
   0   1     2      3       4       5 ...
  -1   0     1      0      -7     -28 ...
  -2  -1     0    -17    -118    -513 ...
  -3   0    17      0    -399   -2800 ...
  -4   7   118    399       0   -7849 ...
  -5  28   513   2800    7849       0 ...
  ...
The start of the sequence as a triangular array, read by rows (i.e., descending antidiagonals of T(n,k)), is as follows:
  0;
  1,  -1;
  2,   0,   -2;
  3,   1,   -1, -3;
  4,   0,    0,  0,  -4;
  5,  -7,  -17, 17,   7, -5;
  6, -28, -118,  0, 118, 28, -6;
  ...
In the above triangle, row number m contains m numbers: m^1 - 1^m, (m-1)^2 - 2^(m-1), ..., 1^m - m^1.

Boris Putievskiy, Rows n = 1..76 of triangle, flattened

Cf. A002260, A004736, A051128, A051129, A055651, A156353.

matrix(9, 9, n, k, k^n - n^k) \\ Michel Marcus, Oct 04 2019

t=int((math.sqrt(8*n-7) - 1)/ 2)
m=((t*t+3*t+4)/2-n)**(n-t*(t+1)/2)-(n-t*(t+1)/2)**((t*t+3*t+4)/2-n)

As a linear array, the sequence is a(n) = A004736(n)^A002260(n) - A002260(n)^A004736(n) or

a(n) = ((t*t + 3*t + 4)/2 - n)^(n - t*(t + 1)/2) - (n - t*(t + 1)/2)^((t*t + 3*t + 4)/2 - n) where t = floor((-1 + sqrt(8*n - 7))/2).

A082754 Triangle read by rows: T(n, k) = abs(n^k-k^n), 1<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 0, 17, 0, 4, 7, 118, 399, 0, 5, 28, 513, 2800, 7849, 0, 6, 79, 1844, 13983, 61318, 162287, 0, 7, 192, 6049, 61440, 357857, 1417472, 3667649, 0, 8, 431, 18954, 255583, 1894076, 9546255, 35570638, 91171007, 0, 9, 924, 58049, 1038576

Offset: 1

Amarnath Murthy, Apr 17 2003

Aside from n = k, the only zero in this triangle corresponds to a(4, 2). - Alonso del Arte, Jul 09 2009

			Triangle begins:
  0
  1 0
  2 1 0
  3 0 17 0
  4 7 118 399 0
  5 28 513 2800 7849 0
  ...

Cf. A055651.

a[n_, k_] := a[n, k] = n^k - k^n; ColumnForm[Table[Abs[a[n, k]], {n, 10}, {k, n}], Center] (* Alonso del Arte, Jul 09 2009 *)

More terms from David Wasserman, Oct 04 2004

A062283 Square array read by descending antidiagonals: T(n,k) = floor(n^k/k^n).

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 1, 4, 0, 1, 1, 1, 5, 0, 1, 1, 0, 0, 6, 0, 1, 1, 1, 0, 0, 7, 0, 2, 3, 1, 0, 0, 0, 8, 0, 4, 6, 3, 1, 0, 0, 0, 9, 0, 6, 12, 6, 2, 0, 0, 0, 0, 10, 0, 10, 27, 16, 4, 1, 0, 0, 0, 0, 11, 0, 16, 59, 39, 11, 2, 0, 0, 0, 0, 0, 12, 0, 28, 133, 104, 33, 6, 1, 0, 0, 0, 0, 0, 13

Offset: 1

Henry Bottomley, Jul 02 2001

			T(3,2) = floor(3^2/2^3) = floor(9/8) = 1.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals).

Cf. A055651, A060505.

Table[Floor[k^(n-k+1)/(n-k+1)^k], {n, 15}, {k, n}] (* Paolo Xausa, May 06 2024 *)

cp's OEIS Frontend

A007925 a(n) = n^(n+1) - (n+1)^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A123206 Primes of the form x^y - y^x, for x,y > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Views

Author

Keywords

Crossrefs

Formula

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A220417 Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A082754 Triangle read by rows: T(n, k) = abs(n^k-k^n), 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A062283 Square array read by descending antidiagonals: T(n,k) = floor(n^k/k^n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica