A072180 -id:A072180 - cp's OEIS frontend

A024012 a(n) = 2^n - n^2.

1, 1, 0, -1, 0, 7, 28, 79, 192, 431, 924, 1927, 3952, 8023, 16188, 32543, 65280, 130783, 261820, 523927, 1048176, 2096711, 4193820, 8388079, 16776640, 33553807, 67108188, 134216999, 268434672, 536870071, 1073740924, 2147482687, 4294966272, 8589933503, 17179868028, 34359737143

Offset: 0

N. J. A. Sloane

The sequence 2^(n-2) - (n-2)^2, n=7,8,... enumerates the number of non-isomorphic sequences of length n, with entries from {1,2,3} and no two adjacent entries the same, that contain each of the thirteen rankings of three players (111, 121, 112, 211, 122, 212, 221, 123, 132, 213, 231, 312, 321) as embedded order isomorphic subsequences. See the arXiv paper below for proof. If n=7, these sequences are 1213121, 1213212, 1231213, 1231231,1231321, 1232123, and 1232132, and for each case, there are 3!=6 isomorphs. - Anant Godbole, Feb 20 2013

GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Anant Godbole and Martha Liendo, Waiting time distribution for the emergence of superpatterns, arxiv 1302.4668 [math.PR], 2013.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A072180 (2^n - n^2 is prime), A075896 (primes of the form 2^n - n^2), A099481 (2^n - n^2 is a semiprime), A099482 (semiprimes of the form 2^n - n^2).

[2^n-n^2: n in [0..30]]; // Vincenzo Librandi, Apr 29 2011

seq(2^n-n^2, n=0..35); # Zerinvary Lajos, Jul 01 2007

CoefficientList[Series[(1 - 4*x + 4*x^2 + x^3)/((1 - x)^3(1 - 2x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 13 2012 *)
Table[2^n - n^2, {n, 0, 39}] (* Alonso del Arte, Dec 16 2012 *)

A024012(n):=2^n-n^2$ makelist(A024012(n),n,0,20); /* Martin Ettl, Dec 18 2012 */

a(n)=2^n-n^2 \\ Charles R Greathouse IV, Apr 17 2012

G.f.: (1 - 4*x + 4*x^2 + x^3)/((1 - 2*x)*(1 - x)^3). - Vincenzo Librandi, Jul 13 2012

a(n) = 5*a(n - 1) - 9*a(n - 2) + 7*a(n - 3) - 2*a(n - 4). - Vincenzo Librandi, Jul 13 2012

More terms from Hugo Pfoertner, Oct 18 2004

A075896 Primes of the form 2^k - k^2.

Original entry on oeis.org

7, 79, 431, 130783, 523927, 2251799813682647, 9007199254738183, 2417851639229258349405791, 9671406556917033397642519, 664613997892457936451903530140158127, 784637716923335095479473677900958302012794430558004278391

Offset: 1

Zak Seidov, Oct 17 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..20

Primes in A024012.

Cf. A061119, A072180.

[a: n in [1..200] | IsPrime(a) where a is 2^n-n^2]; // Vincenzo Librandi, Dec 07 2011

Select[Table[2^n - n^2, {n, 500}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

for(n=2,10^7,if(isprime(2^n-n^2),print1(2^n-n^2",")))

a(n) = A024012(A072180(n)). - Elmo R. Oliveira, Feb 18 2025

More terms from Ralf Stephan, Mar 30 2003

A122003 Numbers n such that A024152(n) = 12^n - n^12 is prime.

Original entry on oeis.org

1, 13, 25, 325, 833, 2087, 29773

Offset: 1

Alexander Adamchuk, Sep 12 2006

Corresponding primes of the form A024152[n] = 12^n - n^12 are {11,83695120256591,953962166381085484825907807,...}.

a(8) > 50000. - Michael S. Branicky, Oct 01 2024

Cf. A024152, A072180, A109387.

Do[f=12^n-n^12;If[PrimeQ[f],Print[{n,f}]],{n,1,833}]

is(n)=isprime(12^n-n^12) \\ Charles R Greathouse IV, Feb 17 2017

a(6) from Robert G. Wilson v, Sep 14 2006

a(7) from Donovan Johnson, Feb 26 2008

A128447 Numbers k such that absolute value of 7^k - k^7 is prime.

Original entry on oeis.org

2, 6, 20, 24, 18582, 20366

Offset: 1

Alexander Adamchuk, Mar 03 2007

From the Lifchitz link: 116240, 188858, and 230492 are also terms. - Michael S. Branicky, Jul 29 2024

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128448, A128449, A128450, A128451, A122003, A128453, A128454.

lst={};k=7;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=isprime(abs(7^n-n^7)) \\ Charles R Greathouse IV, Feb 17 2017

a(5) and a(6) from Donovan Johnson, Mar 03 2008

A128448 Numbers k such that 8^k - k^8 is prime.

Original entry on oeis.org

1, 11, 89, 201, 977, 1351, 3869, 60681

Offset: 1

Alexander Adamchuk, Mar 03 2007

From the Lifchitz link: 60681 and 85349 are also in this sequence. - Robert Price, Mar 27 2019

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.

lst={};k=8;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=isprime(8^n-n^8) \\ Charles R Greathouse IV, Feb 17 2017

a(6) and a(7) from Donovan Johnson, Feb 26 2008

a(8) confirmed (no intervening terms) by Michael S. Branicky, Jul 29 2024

A128449 Numbers k such that absolute value of 9^k - k^9 is prime.

Original entry on oeis.org

2, 10, 50, 7900, 18494, 23840, 36838

Offset: 1

Alexander Adamchuk, Mar 03 2007

Three larger terms 18494, 23840 and 36838 found by Donovan Johnson, Jul-Aug 2005.

From the Lifchitz link: 83024 is also a term. - Michael S. Branicky, Jul 29 2024

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128450, A128451, A122003, A128453, A128454.

lst={};k=9;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=isprime(abs(9^n-n^9)) \\ Charles R Greathouse IV, Feb 17 2017

a(4)-a(6) from Ryan Propper, Feb 22 2008

a(7) from Donovan Johnson, Feb 26 2008

A128450 Numbers k such that absolute value of 10^k - k^10 is prime.

Original entry on oeis.org

3, 9, 273, 399

Offset: 1

Alexander Adamchuk, Mar 03 2007

a(5) > 10^5 if it exists. - Michael S. Branicky, Nov 27 2024

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128451, A122003, A128453, A128454.

[n: n in [0..500]| IsPrime(10^n-n^10)]; // Vincenzo Librandi, Apr 01 2015

lst={};k=10;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0, 10000], PrimeQ[10^# - #^10] &] (* Vincenzo Librandi, Apr 01 2015 *)

is(n)=isprime(abs(10^n-n^10)) \\ Charles R Greathouse IV, Feb 17 2017

A128451 Numbers k such that the absolute value of 11^k - k^11 is prime.

Original entry on oeis.org

8, 14, 80, 212, 230, 1352, 13674, 16094, 44772

Offset: 1

Alexander Adamchuk, Mar 03 2007

Two larger terms 13674 and 16094 found by Donovan Johnson, Jul 2005.

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450, A122003, A128453.

lst={};k=11;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=ispseudoprime(abs(11^n-n^11)) \\ Charles R Greathouse IV, Feb 17 2017

a(6)-a(8) from Donovan Johnson, Feb 26 2008

a(9) discovered by Serge Batalov, entered by Robert Price, Apr 11 2019

A123206 Primes of the form x^y - y^x, for x,y > 1.

Original entry on oeis.org

7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249

Offset: 1

Alexander Adamchuk, Oct 04 2006

nonn

These are the primes in A045575, numbers of the form x^y - y^x, for x,y > 1. This includes all primes from A122735, smallest prime of the form (n^k - k^n) for k>1.

If y=1 was allowed, any prime p could be obtained for x=p+1. This motivates to consider sequence A243100 of primes of the form x^(y+1)-y^x. - M. F. Hasler, Aug 19 2014

			The primes 6102977801 and 1490116119372884249 are of the form 5^y-y^5 (for y=14 and y=26) and therefore members of this sequence. The next larger primes of this form would have y > 4500 and would be much too large to be included. - _M. F. Hasler_, Aug 19 2014

T. D. Noe, Table of n, a(n) for n=1..101 (terms < 10^400)
H. Lifchitz & R. Lifchitz, PRP of the form x^y-y^x on primenumbers.net.

Cf. A045575, A122735, A078202, A082754, A055651, A094133.
A163319 is the subsequences for fixed x=3, A243114 for x=6.
Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.

N:= 10^100: # to get all terms <= N
A:= NULL:
for x from 2 while x^(x+1) - (x+1)^x <= N do
   for y from x+1 do
      z:= x^y - y^x;
      if z > N then break
      elif z > 0 and isprime(z) then A:=A, z;
      fi
od od:
{A}; # Robert Israel, Aug 29 2014

Take[Select[Intersection[Flatten[Table[Abs[x^y-y^x],{x,2,120},{y,2,120}]]],PrimeQ[ # ]&],25]
nn=10^50; n=1; t=Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num0&&PrimeQ[#]&]],nn]] (* Harvey P. Dale, Nov 23 2013 *)

a=[];for(S=1,199,for(x=2,S-2,ispseudoprime(p=x^(y=S-x)-y^x)&&a=concat(a,p)));Set(a) \\ May be incomplete in the upper range of values, i.e., beyond a given S=x+y, a larger S may yield a smaller prime (for small x). - M. F. Hasler, Aug 19 2014

A064539 Numbers n such that 2^n + n^2 is prime.

Original entry on oeis.org

1, 3, 9, 15, 21, 33, 2007, 2127, 3759, 29355, 34653, 57285, 99069, 1933695

Offset: 1

Jason Earls, Oct 16 2001

Values 2^2007+2007^2, 2^2127+2127^2, 2^3759+3759^2 were proved prime with Primo.
n is always an odd multiple of 3 (except for the first term), i.e., a(n) is a subsequence of A016945. - Lekraj Beedassy, Jan 01 2007
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 165 pp. 30, 160, Ellipses Paris 2004.

Henri Lifchitz, Renaud Lifchitz, PRP Top Records. 2^n+n^2.

Cf. A001580, A016945, A061119, A072180, A094133.

for(n=1,5000, if(isprime(2^n+n^2),print(n)))

a(10)-a(13) from Hugo Pfoertner, Jun 26 2004

a(14) from Ryan Propper, May 11 2023. n=1933695 corresponds to a probable prime with 582101 digits, and was PRP tested with PFGW.

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A024012 a(n) = 2^n - n^2.

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A075896 Primes of the form 2^k - k^2.

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A122003 Numbers n such that A024152(n) = 12^n - n^12 is prime.

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A128447 Numbers k such that absolute value of 7^k - k^7 is prime.

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A128448 Numbers k such that 8^k - k^8 is prime.

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A128449 Numbers k such that absolute value of 9^k - k^9 is prime.

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A128450 Numbers k such that absolute value of 10^k - k^10 is prime.

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