A099498 -id:A099498 - 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

A099497 Numbers k such that A007925(k) = k^(k+1) - (k+1)^k is a semiprime.

Original entry on oeis.org

5, 7, 8, 11, 17, 18, 21, 23, 25, 27, 32, 47, 51, 56, 59, 165

Offset: 1

Hugo Pfoertner, Oct 19 2004, Aug 13 2007

a(15)=59 confirmed by the factorization of 59^60 - 60^59, which is the product of the 52-digit prime 1994803969065168661575061125592557043358338451845483 and the 55-digit prime 8529249434913526091880095870250840825853220069057672947.

The next term is >= 182. - Hugo Pfoertner, Jul 18 2019

			a(1) = 5 because 5^6 - 6^5 = 7849 = 47*167 is a semiprime.
a(1) = 5 because 5^6 - 6^5 = 47*167
a(2) = 7 because 7^8 - 8^7 = 23*159463
a(3) = 8 because 8^9 - 9^8 = 257*354751
a(4) = 11 because 11^12 - 12^11 = 33479*71549927
a(5) = 17 because 17^18 - 18^17 = 443881*26757560905578361
a(6) = 18 because 18^19 - 19^18 = 100417*6015993258685545623
a(7) = 21 because 21^22 - 22^21 = 10745792197529*9973660056412561
a(8) = 23 because 23^24 - 24^23 = 92798617729*4576344458074395243073
a(9) = 25 because 25^26 - 26^25 = 1627*1219220786258356172077730898121187
a(10) = 27 because 27^28 - 28^27 = 12298336501553*877252504725615101634783073
a(11) = 32 because 32^33 - 33^32 = 3506869732968391733353*12220478717670771804763962407
a(12) = 47 because 47^48 - 48^47 = 11*15621013371424880252957237277868559270462038147831682437840584991339231377934499
a(13) = 51 because 51^52 - 52^51 = 10562756058978342869988055703171*5575962824795589360993690554534422732411612977322491058843
a(14) = 56 because 56^57 - 57^56 = 5*843980334169667457302970806376511482920948635540290643213973523914715036518308339240201775858865907
a(15) = 59 because 59^60 - 60^59 = 1994803969065168661575061125592557043358338451845483*8529249434913526091880095870250840825853220069057672947
a(16) = 165 because 165^166 - 166^165 = 7633959407*16307690786821361595026621717879347561301150483781862339651556401266189322630373265190696672506475741217560239791446654891805648807872536646884416611251422684856600732984767987061831649144878649678190762809385448362714901584206533854093359279076584767352259587745683931159999248465944943129517543272252180930134912057221968601458271001580745436226192252814407

Dario Alpern, Factorization using the Elliptic Curve Method.
factordb.com, Status of 182^183-183^182.
Jason Papadopoulos, MSIEVE: A Library for Factoring Large Integers.

Cf. A007925 (n^(n+1)-(n+1)^n), A072179 (k^(k+1)-(k+1)^k is prime), A099498 (semiprimes of the form k^(k+1)-(k+1)^k).

165 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 12 2008

A099499 Primes of the form A007925(n)=n^(n+1)-(n+1)^n.

Original entry on oeis.org

17, 162287, 2486784401, 83695120256591, 84721522804414816904952398305908708011513455440403306207160333176150520399

Offset: 1

Hugo Pfoertner, Oct 19 2004

nonn

The next term a(6)=883^884-884^883 has 2605 decimal digits and is too large to display.

			a(2)=162287 because A007925(A072179(2))=6^7-7^6=162287 is prime.

Cf. A007925 n^(n+1)-(n+1)^n, A072179 n^(n+1)-(n+1)^n is prime, A099497 n^(n+1)-(n+1)^n is a semiprime, A099498 semiprimes of the form n^(n+1)-(n+1)^n.

[a: n in [0..50] | IsPrime(a) where a is n^(n+1)-(n+1)^n ]; // Vincenzo Librandi, Jul 18 2012

Select[Table[n^(n+1)-(n+1)^n,{n,0,1000}],PrimeQ] (* Vincenzo Librandi, Jul 18 2012 *)

cp's OEIS Frontend

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

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A099497 Numbers k such that A007925(k) = k^(k+1) - (k+1)^k is a semiprime.

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A099499 Primes of the form A007925(n)=n^(n+1)-(n+1)^n.

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