A007925 -id:A007925 - cp's OEIS frontend

A099498 Semiprimes of the form A007925(n) = n^(n+1)-(n+1)^n.

7849, 3667649, 91171007, 2395420006033, 11877172892329028459041, 604107995057426434824791, 107174878415004743976428761769, 424678439961073471604787362241217, 1983672219242345491970468171243171249, 10788746499945827829225142589096882612369, 42855626937384013751014398588294858582343260060671

Offset: 1

Hugo Pfoertner, Oct 19 2004

nonn

			a(1)=7849 because 5^6-6^5=7849=47*167 is a semiprime.

Cf. A007925 n^(n+1)-(n+1)^n, A072179 n^(n+1)-(n+1)^n is prime, A099499 primes of the form n^(n+1)-(n+1)^n, A099497 n^(n+1)-(n+1)^n is a semiprime.

IsSemiprime:=func; [s: n in [3..30] | IsSemiprime(s) where s is n^(n+1)-(n+1)^n]; // Vincenzo Librandi, Sep 21 2012

Select[Table[n^(n + 1) - (n + 1)^n, {n, 30}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)

a(9)-a(11) from Vincenzo Librandi, Sep 21 2012

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 *)

A145328 Partial sums of A007925, starting at n=1.

Original entry on oeis.org

-1, -2, 15, 414, 8263, 170550, 3838199, 95009206, 2581793607, 76644369006, 2472064375039, 86167184631630, 3229828797076775, 129604998329498374, 5545091849435542023, 252031805153121499398, 12129204697482149958439

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 08 2008

sign

lst={};s=0;Do[s+=n^(n+1)-(n+1)^n;AppendTo[lst,s],{n,5!}];lst
Accumulate[Table[n^(n+1)-(n+1)^n,{n,20}]] (* Harvey P. Dale, Aug 26 2012 *)

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

A174380 Smallest prime factors of numbers of the form (n-1)^n - n^(n-1) A007925.

Original entry on oeis.org

17, 3, 47, 162287, 23, 257, 2486784401, 3, 33479, 83695120256591, 5, 67, 7, 3, 443881, 100417, 859, 79, 10745792197529, 3, 92798617729, 67, 1627, 11, 12298336501553, 3, 19, 241, 167, 3506869732968391733353, 5, 3, 47, 5, 317

Offset: 4

Torbjorn Alm (talm(AT)tele2.se), Mar 17 2010

nonn

For n = 1 to 3 no prime factors in A007925(n).

A007925, A174379

Table[FactorInteger[(n-1)^n-n^(n-1)][[1,1]],{n,4,40}] (* Harvey P. Dale, May 25 2011 *)

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

Original entry on oeis.org

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)

A046065 a(n) = n^(n+2) - (n+2)^n.

Original entry on oeis.org

-1, -2, 0, 118, 2800, 61318, 1417472, 35570638, 973741824, 29023111918, 938082635776, 32730551749894, 1227224552173568, 49239697945731382, 2105895743771443200, 95663702284183543582, 4600926951773050961920, 233592048827366522661214

Offset: 0

N. J. A. Sloane

sign

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

Cf. A007925 (n^(n+1) - (n+1)^n).

[n^(n+2) -(n+2)^n: n in [0..40]]; // G. C. Greubel, Jul 14 2021

a[n_]:=n^(n+2)-(n+2)^n;lst={};Do[AppendTo[lst,a[n]],{n,0,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 20 2008 *)
Table[n^(n+2)-(n+2)^n,{n,0,20}] (* Harvey P. Dale, May 31 2025 *)

[n^(n+2) -(n+2)^n for n in (0..40)] # G. C. Greubel, Jul 14 2021

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

A055651 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

0, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 2, 0, -2, -1, 1, 3, 1, -1, -3, -1, 1, 4, 0, 0, 0, -4, -1, 1, 5, -7, -17, 17, 7, -5, -1, 1, 6, -28, -118, 0, 118, 28, -6, -1, 1, 7, -79, -513, -399, 399, 513, 79, -7, -1, 1, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, -1, 1, 9, -431

Offset: 0

Henry Bottomley, Jun 07 2000

Rows A000012 (offset), A023443, A024012, A024026, A024040 and diagonals A000004, A007925, A046065, A055652.

Title corrected by Sean A. Irvine, Mar 30 2022

A178922 a(n) = (n+1)^n - n^(n-1) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 7, 55, 561, 7151, 109873, 1979503, 40949569, 956953279, 24937424601, 717070946087, 22555076751793, 770416688131663, 28399211252136481, 1123728578581456351, 47508270371060021505, 2137250367863029663487, 101941438738172545000873, 5138752649702088758467159

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 29 2010

nonn

Cf. A000169, A007925, A084363, A152917.

a:= n-> (f-> f(n+1)-f(n))(n-> `if`(n=0, 0, n^(n-1))):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 26 2020

Mathematica
```
Table[(n+1)^n-n^(n-1),{n,25}]
```

A178922[n]:=(n+1)^n-n^(n-1)$ makelist(A178922[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

vector(100, n, (n+1)^n - n^(n-1)) \\ Altug Alkan, Oct 19 2015

a(n) = A152917(n+1) - A152917(n). - Alexei Kourbatov, Oct 19 2015

E.g.f.: W(-x) - W(-x)/(x*(1+W(-x))) where W is the Lambert W function. - Robert Israel, Oct 19 2015

a(0)=1 prepended and definition adapted by Alois P. Heinz, Feb 26 2020

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

cp's OEIS Frontend

A099498 Semiprimes of the form A007925(n) = n^(n+1)-(n+1)^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A145328 Partial sums of A007925, starting at n=1.

Views

Author

Keywords

Programs

Mathematica

Extensions

A174380 Smallest prime factors of numbers of the form (n-1)^n - n^(n-1) A007925.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A046065 a(n) = n^(n+2) - (n+2)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A055651 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

Extensions

A178922 a(n) = (n+1)^n - n^(n-1) for n > 0, a(0) = 1.

Views

Author

Keywords

Crossrefs