A068146 -id:A068146 - cp's OEIS frontend

A007781 a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

1, 3, 23, 229, 2869, 43531, 776887, 15953673, 370643273, 9612579511, 275311670611, 8630788777645, 293959006143997, 10809131718965763, 426781883555301359, 18008850183328692241, 808793517812627212561

Offset: 0

Peter McCormack (peter.mccormack(AT)its.csiro.au)

(12n^2 + 6n + 1)^2 divides a(6n+1), where (12n^2 + 6n + 1) = (2n+1)^3 - (2n)^3 = A127854(n) = A003215(2n) are the hex (or centered hexagonal) numbers. The prime numbers of the form 12n^2 + 6n + 1 belong to A002407. - Alexander Adamchuk, Apr 09 2007

			a(14) = 10809131718965763 = 3 * 61^2 * 968299894201.

Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

Doug Bell, Table of n, a(n) for n = 0..100
Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
Ronald K. Hoeflin, Mega Test. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Power Difference Prime.

Cf. A000312, A068146, A068954, A068955, A068956, A068957.

Cf. A002407, A003215, A127854.

[1] cat [(n+1)^(n+1)-n^n: n in [1..20]]; // Vincenzo Librandi, Aug 19 2015

seq( `if`(n=0,1,(n+1)^(n+1) -n^n), n=0..20); # G. C. Greubel, Mar 05 2020

Join[{1},Table[(n+1)^(n+1)-n^n,{n,20}]]  (* Harvey P. Dale, Feb 09 2011 *)
Differences[Table[n^n,{n,0,20}]] (* Charles R Greathouse IV, Feb 09 2011 *)

first(m)=vector(m,i,i--;(i+1)^(i+1) - i^i) /* Anders Hellström, Aug 18 2015 */

[1]+[(n+1)^(n+1) -n^n for n in (1..20)] # G. C. Greubel, Mar 05 2020

a(n) = A000312(n+1) - A000312(n) for n>0, a(0) = 1.
a(n) = abs(discriminant(x^(n+1)-x+1)).
E.g.f.: W(-x)/(1+W(-x)) - W(-x)/((1+W(-x))^3*x) where W is the Lambert W function. - Robert Israel, Aug 19 2015
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

A124076 Numbers of the form Abs[m^m - n^n], where integers m,n>0.

Original entry on oeis.org

0, 3, 23, 26, 229, 252, 255, 2869, 3098, 3121, 3124, 43531, 46400, 46629, 46652, 46655, 776887, 820418, 823287, 823516, 823539, 823542, 15953673, 16730560, 16774091, 16776960, 16777189, 16777212, 16777215, 370643273, 386596946

Offset: 1

Alexander Adamchuk, Nov 24 2006

nonn

Prime a(n) are listed in A068146(n) = {3, 23, 229, 3121, 776887, 16774091, 275311670611, ...}.

Cf. A066846 = Integers of the form m^m + n^n, m and n = positive integers. Cf. A068146 = Primes of the form a^a - b^b.

Take[Intersection[Flatten[Table[Abs[x^x-y^y], {x, 1, 12}, {y, 1, 12}]]], 50]

A162408 Solutions x to the equation x^x-y^y = some prime number for any y.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 11, 13, 15, 17, 19, 23, 26, 30, 42, 47, 53, 65, 73, 77, 84, 92, 100, 101, 106, 110, 116, 120, 122, 122, 124, 133, 137, 163, 167, 173, 173

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 02 2009

nonn

These are the numbers a in the definition of A068146.
If there are two solutions, like with (x,y) = (17,12) and (x,y) = (17,16) with
the same x, only one instance of x is placed into the sequence, so there is no
1-to-1 correspondence with terms in A068146.
The corresponding set of y contains at least the numbers 1 to 6, 10, 12, 14, 16, 17, 19, 20, 22 etc

			Triples (x,y,prime) are (2,1,3), (3,2,23), (4,3,229), (5,2,3121), (7,6,776887),
(8,5,16774091), (11,10,275311670611), (13,6,302875106545597), (15,4,437893890380859119),
(17,12,827240252970236315921), (17,16,808793517812627212561) etc

Cf. A068146

f[a_,b_]:=a^a-b^b; lst={};Do[Do[If[a>b,p=f[a,b]];If[PrimeQ[p],AppendTo[lst, a]],{b,4*4!}],{a,5*4!}];Union[lst]

Edited and extended by R. J. Mathar, Sep 16 2009

cp's OEIS Frontend

A007781 a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

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A124076 Numbers of the form Abs[m^m - n^n], where integers m,n>0.

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A162408 Solutions x to the equation x^x-y^y = some prime number for any y.

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