A051442 -id:A051442 - cp's OEIS frontend

A125074 A051442[p-2]/p for p = Prime[n]>2.

1, 29, 3343, 407889491, 298572057493, 454195874136455153, 855210316410864290683, 6009294624226370518329498431, 453664960868214198206218533253007296517

Offset: 2

Alexander Adamchuk, Nov 18 2006

nonn

A051442(n) = n^(n+1)+(n+1)^n. Odd prime p divides A051442(p-2).

Cf. A051442.

Table[((Prime[n]-2)^(Prime[n]-1)+(Prime[n]-1)^(Prime[n]-2))/Prime[n],{n,2,15}]

a(n) = A051442[ Prime[n] - 2 ]/Prime[n] for n>1.

A145329 Partial sums of A051442, starting at n=1.

Original entry on oeis.org

3, 20, 165, 1814, 25215, 422800, 8284753, 185549202, 4672333603, 130609758204, 4012046505613, 134303337007166, 4865394495960599, 189626416079163448, 7910956276398901049, 351720053331418595386, 16600716065137840118363

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 08 2008

nonn

lst={};s=0;Do[s+=n^(n+1)+(n+1)^n;AppendTo[lst,s],{n,4!}];lst
Accumulate[Table[n^(n+1)+(n+1)^n,{n,20}]] (* Harvey P. Dale, Oct 02 2018 *)

s+=n^(n+1)+(n+1)^n.

Edited by N. J. A. Sloane, Oct 09 2008

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

Original entry on oeis.org

-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

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]

A073499 Numbers k such that k^(k+1) + (k+1)^k is prime.

Original entry on oeis.org

1, 2, 80, 342, 848, 1194, 2658, 4790, 9376

Offset: 1

Rick L. Shepherd, Aug 05 2002

a(10) > 20000. - Michael S. Branicky, Apr 13 2025

			1^2 + 2^1 = 3 and 2^3 + 3^2 = 17 are the primes corresponding to the first two terms. The next five terms correspond to primes of 155, 870, 2487, 3678 and 9106 decimal digits.

P. Leyland, Primes and Strong Pseudoprimes of the form x^y + y^x - The numbers corresponding to the first 7 terms had been proved prime by 2005.

Cf. A072179, A051442, A094133.

Do[ If[ PrimeQ[n^(n + 1) + (n + 1)^n], Print[n]], {n, 1, 1650}]

for(n=1,1650, if(isprime((n^(n+1))+((n+1)^n)), print1(n,",")))

Edited by Robert G. Wilson v, Aug 08 2002
a(7) from Charles R Greathouse IV, Jan 13 2012
a(8) from Charles R Greathouse IV, Jan 17 2012
a(9) found by Alexander Adamchuk, Apr 09 2007 and shown to be a(9) by Charles R Greathouse IV, Jan 27 2012

A051489 a(n) = n^(n+2) + (n+2)^n.

Original entry on oeis.org

1, 4, 32, 368, 5392, 94932, 1941760, 45136576, 1173741824, 33739007300, 1061917364224, 36314872537968, 1340612376924160, 53132088082450132, 2250010931847299072, 101388548387203175168, 4843806013966239465472

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A007925, A046065, A051442. - Vladimir Joseph Stephan Orlovsky, Jan 23 2009

[n^(n+2) + (n+2)^n: n in [0..30]]; // G. C. Greubel, Jul 14 2021

Table[n^(n+2)+(n+2)^n,{n,0,20}] (* Harvey P. Dale, Jul 28 2025 *)

[n^(n+2) + (n+2)^n for n in (0..30)] # G. C. Greubel, Jul 14 2021

A051443 a(n) = n^(n+1)*(n+1)^n.

Original entry on oeis.org

0, 2, 72, 5184, 640000, 121500000, 32934190464, 12089663946752, 5777633090469888, 3486784401000000000, 2593742460100000000000, 2331878554708454877954048, 2492736806448711465154117632, 3125153805191532199063557103616, 4541487905530112153400000000000000

Offset: 0

N. J. A. Sloane

Cf. A007925, A051442.

Table[n^(n+1)*(n+1)^n,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2010 *)

A051443[n]:=n^(n+1)*(n+1)^n$ makelist(A051443[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

A155539 a(n) = n^(n+3) + (n+3)^n.

Original entry on oeis.org

1, 5, 57, 945, 18785, 423393, 10609137, 292475249, 8804293473, 287589316833, 10137858491849, 383799398752905, 15536767912476993, 669920208810550337, 30659724555890596833, 1484638520651877849057, 75846305139481944586817

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

nonn

1^4 + 4^1 = 5, 2^5 + 5^2 = 57, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A046065, A051442, A007925, A051489, A153217, A076980.

[n^(n+3)+(n+3)^n: n in [0..20] ]; // Vincenzo Librandi, Aug 25 2011

lst={};Do[m=n+3;q=n^m+m^n;AppendTo[lst,q],{n,0,4!}];lst
Table[n^(n+3)+(n+3)^n,{n,0,20}] (* Harvey P. Dale, Aug 18 2024 *)

Offset corrected by Arkadiusz Wesolowski, Aug 24 2011

A258389 a(n) = (n^(n+1)-(n-1)^n) + ((n+1)^n-n^(n-1)).

Original entry on oeis.org

2, 14, 128, 1504, 21752, 374184, 7464368, 169402496, 4309519952, 121450640200, 3755499322808, 126409853754144, 4600799868451880, 180029930424249416, 7536568838736534752, 336087767194699956736, 15905186914751401828640, 796113699641442496367496

Offset: 1

Daniel Suteu, May 28 2015

nonn

			For a(3) = (3^(3+1)-(3-1)^3) + ((3+1)^3-3^(3-1)) = (3^4 - 2^3) + (4^3 - 3^2) = 128.

Daniel Suteu, Table of n, a(n) for n = 1..20

Cf. A051442, A084363, A178922.

[(n^(n+1)-(n-1)^n) + ((n+1)^n-n^(n-1)): n in [1..20]]; // Vincenzo Librandi, May 29 2015

Array[#^(# + 1) - (# - 1)^# + ((# + 1)^# - #^(# - 1)) &, 20] (* Vincenzo Librandi, May 29 2015 *)

func a(n) {
     ((n+1)**n - n**(n-1)) -
     ((n-1)**n - n**(n+1))
};
1.to(Math.inf).each { |n|
    say a(n);
};

a(n) = (n^(n+1)-(n-1)^n) + ((n+1)^n-n^(n-1)) = A084363(n) + A178922(n).

a(n) = A051442(n) - A051442(n-1). - Mathew Englander, Jul 08 2020

A161473 Primes of the form k^(k+1)+(k+1)^k.

Original entry on oeis.org

3, 17, 14612087592038378751152858509524512533536096028044190178822935218486730194880516808459166772134240378240755073828170296740373082348622309614668344831750401

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 10 2009

The associated k are in A073499. [R. J. Mathar, Jun 12 2009]

			2^1+1^2=3. 2^3+3^2=17.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..4

lst={};Do[p=n^(n+1)+(n+1)^n;If[PrimeQ[p],AppendTo[lst,p]],{n,2*5!}];lst

A051442 INTERSECT A000040. [R. J. Mathar, Jun 12 2009]

Definition simplified by R. J. Mathar, Jun 12 2009

cp's OEIS Frontend

A125074 A051442[p-2]/p for p = Prime[n]>2.

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A145329 Partial sums of A051442, starting at n=1.

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

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

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A073499 Numbers k such that k^(k+1) + (k+1)^k is prime.

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A051489 a(n) = n^(n+2) + (n+2)^n.

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Sage

A051443 a(n) = n^(n+1)*(n+1)^n.

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Maxima

A155539 a(n) = n^(n+3) + (n+3)^n.

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