A007925 -id:A007925

A099498

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

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..11.

EXAMPLE

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

MATHEMATICA

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

PROG

(Magma) IsSemiprime:=func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [3..30] | IsSemiprime(s) where s is n^(n+1)-(n+1)^n]; // Vincenzo Librandi, Sep 21 2012

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Oct 19 2004

EXTENSIONS

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

STATUS

approved

A099497

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..16.

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.

EXAMPLE

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

CROSSREFS

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).

KEYWORD

nonn,hard,more

AUTHOR

Hugo Pfoertner, Oct 19 2004, Aug 13 2007

EXTENSIONS

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

STATUS

approved

A099499

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

+20
1

17, 162287, 2486784401, 83695120256591, 84721522804414816904952398305908708011513455440403306207160333176150520399

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..5.

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Oct 19 2004

STATUS

approved

A145328

Partial sums of A007925, starting at n=1.

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..17.

MATHEMATICA

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 *)

KEYWORD

sign

AUTHOR

Vladimir Joseph Stephan Orlovsky, Oct 08 2008

EXTENSIONS

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

STATUS

approved

A174380

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,1

COMMENTS

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

LINKS

Table of n, a(n) for n=4..38.

MATHEMATICA

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

CROSSREFS

A007925, A174379

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A051442

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Mathew Englander, Notes on OEIS A051442

FORMULA

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)

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A046065

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

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

KEYWORD

sign

AUTHOR

N. J. A. Sloane

STATUS

approved

A055651

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

+10
7

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,12

LINKS

Table of n, a(n) for n=0..68.

CROSSREFS

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

KEYWORD

easy,sign,tabl

AUTHOR

Henry Bottomley, Jun 07 2000

EXTENSIONS

Title corrected by Sean A. Irvine, Mar 30 2022

STATUS

approved

A178922

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

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

MAPLE

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}]

PROG

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

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

CROSSREFS

Cf. A000169, A007925, A084363, A152917.

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 29 2010

EXTENSIONS

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

STATUS

approved

A051489

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

PROG

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

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved