A051443 -id:A051443 - cp's OEIS frontend

A051442 a(n) = n^(n+1)+(n+1)^n.

1, 3, 17, 145, 1649, 23401, 397585, 7861953, 177264449, 4486784401, 125937424601, 3881436747409, 130291290501553, 4731091158953433, 184761021583202849, 7721329860319737601, 343809097055019694337, 16248996011806421522977

Offset: 0

N. J. A. Sloane

Odd prime p divides a(p-2). For n>1, a(prime(n)-2)/prime(n) = A125074(n) = {1, 29, 3343, 407889491, 298572057493, 454195874136455153, ...}. Prime p divides a((p+5)/2) for p = {19, 23, 61}. - Alexander Adamchuk, Nov 18 2006
From Mathew Englander, Jul 08 2020: (Start)
For all n != 1, a(n) mod 8 = 1.
If n mod 6 = 0, 3, or 5, then a(n) mod 6 = 1. If n mod 6 = 1, then a(n) mod 6 = 3. If n mod 6 = 2 or 4, then a(n) mod 6 = 5.
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, see the Englander link. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mathew Englander, Notes on OEIS A051442

Cf. A007925, A051443, A125074, A076980, A173054, A208506.

[n^(n+1)+(n+1)^n: n in [0..20]]; // Vincenzo Librandi, Jan 12 2012

Table[n^(n+1)+(n+1)^n,{n,0,20}] (* Harvey P. Dale, Oct 02 2018 *)

A051442[n]:=n^(n+1)+(n+1)^n$ makelist(A051442[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n)=(n+1)^n+n^(n+1) \\ Charles R Greathouse IV, Jan 12 2012

a(n) = (n + e + o(1)) * n^n. - Charles R Greathouse IV, Jan 12 2012
From Mathew Englander, Jul 08 2020: (Start)
a(n) = A093898(n+1, n) for n >= 1.
a(n) = a(n-1) + A258389(n) for n >= 1.
a(n) = A007778(n) + A000169(n+1).
(End)

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)

A328217 Numbers k such that k^(k+1)*(k+1)^k + 1 is prime.

Original entry on oeis.org

1, 2, 14, 22, 120

Offset: 1

Juri-Stepan Gerasimov, Oct 08 2019

a(6) > 10^3.
Numbers k such that A051443(k)+1 is prime. - Michel Marcus, Oct 08 2019
a(6) > 10^4, if it exists. - Michael S. Branicky, Aug 24 2024

Cf. A000040, A051443.

[n: n in [1..500] | IsPrime(n^(n+1)*(n+1)^n+1)];

Select[Range[120], PrimeQ[#^(# + 1)*(# + 1)^# + 1] &] (* Amiram Eldar, Oct 08 2019 *)

cp's OEIS Frontend

A051442 a(n) = n^(n+1)+(n+1)^n.

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

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A328217 Numbers k such that k^(k+1)*(k+1)^k + 1 is prime.

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Magma

Mathematica