A075896 -id:A075896 - 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

A072180 Numbers k such that 2^k - k^2 is prime.

Original entry on oeis.org

5, 7, 9, 17, 19, 51, 53, 81, 83, 119, 189, 219, 227, 301, 455, 461, 623, 2037, 2221, 2455, 3547, 5515, 6825, 8303, 9029, 12103, 49989, 55525, 64773, 80307, 119087, 141915, 192023, 205933, 301683, 307407

Offset: 1

Daniel Gronau (Daniel.Gronau(AT)gmx.de), Jun 30 2002

The numbers corresponding to k = 2037, 2221, 3547 and 5515 have been certified prime with Primo. - Rick L. Shepherd, Nov 10 2002
The remaining k's > 1000 correspond only to probable primes.
Certainly k must be odd. Let N(k) = 2^k - k^2. Additional restrictions come from the facts that 7 | N(k) if k is in {2, 4, 5, 6, 10, 15} mod 21 and 17 | N(k) if k is in {31, 57, 61, 71, 107, 109, 113, 131} mod 136. - Daniel Gronau, Jul 06 2002
Henri Lifchitz found the terms > 40000 in 2001 and 119087 in March 2002. - Hugo Pfoertner, Nov 16 2004

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

Cf. A024012, A064539, A075896, A072164.

Do[ If[ PrimeQ[ 2^n - n^2], Print[n]], {n, 1, 22850, 2}]

is(n)=isprime(2^n-n^2) \\ Charles R Greathouse IV, Feb 17 2017

Edited and extended by Robert G. Wilson v, Jul 01 2002
More terms from Hugo Pfoertner, Nov 16 2004
More terms from Henri Lifchitz submitted by Ray Chandler, Mar 02 2007

A061119 Primes in the sequence n^2 + 2^n (A001580).

Original entry on oeis.org

3, 17, 593, 32993, 2097593, 8589935681

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

p and p^2 + 2^p are both prime only for p=3. All positive n satisfy the congruence n=3 (mod 6). - Lekraj Beedassy, Sep 07 2004
For values of n, see A064539. - Lekraj Beedassy, Jan 01 2007
The next term has 605 digits. - Harvey P. Dale, Jul 19 2017

			a(3) = 593 = 2^9 + 9^2.
a(4) = 32993 = 2^15 + 15^2.

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

Subsequence of A094133.

Cf. A001580, A064539, A075896.

Select[Table[n^2+2^n,{n,1000}],PrimeQ] (* Harvey P. Dale, Jul 19 2017 *)

for(n=1,10^7, if(isprime(n^2+2^n),print(n^2+2^n)))

a(n) = A001580(A064539(n)). - Elmo R. Oliveira, Feb 18 2025

More terms from Jason Earls, Aug 09 2001. Next term too large to include.

A099482 Semiprimes of the form 2^k - k^2.

Original entry on oeis.org

1927, 8023, 32543, 2096711, 8388079, 137438952103, 549755812367, 2199023253871, 8796093020359, 140737488353119, 562949953418911, 36028797018960943, 147573952589676408439, 37778931862957161703943

Offset: 1

Hugo Pfoertner, Oct 18 2004

nonn

			a(2) = 8023 because 8023 = 71*113 = 2^13 - 13^2 = 2^A099481(2) - A099481(2)^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..36
Dario Alpern, Factorization using the Elliptic Curve Method.

Cf. A024012 2^n-n^2, A099481 2^k-k^2 is a semiprime, A072180 2^k-k^2 is prime, A075896 primes of the form 2^k-k^2.

Select[Table[2^n - n^2, {n, 100}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)

A099481 Numbers k such that 2^k - k^2 is a semiprime.

Original entry on oeis.org

11, 13, 15, 21, 23, 37, 39, 41, 43, 47, 49, 55, 67, 75, 103, 105, 133, 147, 153, 161, 163, 177, 201, 209, 221, 239, 249, 263, 311, 335, 355, 397, 413, 421, 437, 447, 583, 617, 775, 807

Offset: 1

Hugo Pfoertner, Oct 18 2004

The smaller prime factor of the 125-digit semiprime 2^413 - 413^2 has 40 digits; for the 127-digit semiprime 2^421 - 421^2 the smaller prime factor has 45 digits. The next term is >= 583. - Hugo Pfoertner, Oct 14 2007
The factorization of the 176-decimal-digit composite 2^583 - 583^2 using SNFS in YAFU took 55000 seconds on 4 cores of an i5-2400 CPU @ 3.10GHz. a(38) >= 617. - Hugo Pfoertner, Jul 23 2019
a(41) >= 827. - Hugo Pfoertner, Jul 26 2019

			a(1) = 11 because 2^11 - 11^2 = 1927 = 41*47.

Dario Alpern, Factorization using the Elliptic Curve Method.
factordb, Status of 2^583-583^2 in factordb.com.
factordb, Status of 2^617-617^2 in factordb.com.
factordb, Status of 2^827-827^2 in factordb.com.
YAFU, Automated integer factorization.

Cf. A024012 (2^n-n^2), A099482 (semiprimes of the form 2^n-n^2), A072180 (2^n-n^2 is prime), A075896 (primes of the form 2^n-n^2).

More terms from Hugo Pfoertner, Oct 14 2007

a(37)-a(40) from Hugo Pfoertner, Jul 26 2019

A192515 Number of primes in the range [2^n-n^2, 2^n].

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 10, 11, 15, 15, 16, 16, 18, 19, 20, 21, 23, 23, 31, 24, 34, 28, 27, 35, 32, 41, 38, 46, 45, 38, 44, 36, 49, 51, 43, 61, 33, 48, 58, 42, 62, 67, 59, 63, 70, 57, 63, 73, 68, 85, 74, 75, 73, 77, 86, 85, 74, 94, 89, 83, 89, 94, 93, 97, 102

Offset: 0

Juri-Stepan Gerasimov, Jul 03 2011

nonn

			a(0)=0 because [2^0-0^2, 2^0]=[1, 1],
a(1)=1 because 2 in range [2^1-1^2, 2^1]=[1, 2],
a(2)=2 because 2, 3 in range [2^2-2^2, 2^2]=[0, 4],
a(3)=4 because 2, 3, 5, 7 in range [2^3-3^2, 2^3]=[-1, 8],
a(4)=6 because 2, 3, 5, 7, 11, 13 in range [2^4-4^2, 2^4]=[0, 16],
a(5)=8 because 7, 11, 13, 17, 19, 23, 29, 31 in range [2^5-5^2, 2^5]=[7, 32].

Cf. A007053, A024012, A072180, A075896.

A192515 := proc(n) a := 0 ; for i from 2^n-n^2 to 2^n do if isprime(i) then a := a+1 ; end if; end do; a ; end proc: # R. J. Mathar, Jul 11 2011

Table[Count[Range[2^n - n^2, 2^n], p_ /; PrimeQ@ p], {n, 0, 65}] (* Michael De Vlieger, Apr 03 2016 *)

a(n) = primepi(2^n) - primepi(2^n-n^2) + isprime(2^n-n^2); \\ Michel Marcus, Apr 03 2016

Corrected and extended by R. J. Mathar, Jul 11 2011

A182360 Primes of the form 2^n - n^4.

Original entry on oeis.org

1902671, 33163807, 8588748671, 140737483475647, 562949947656511, 2251799806920047

Offset: 1

Alex Ratushnyak, Apr 26 2012

nonn

Cf. A024014, A075896, A163319.

A222137 Primes of the form 2^p - p^2, where p is prime.

Original entry on oeis.org

7, 79, 130783, 523927, 9007199254738183, 9671406556917033397642519, 215679573337205118357336120696157045389097155380324579848828881942199

Offset: 1

Jaroslav Krizek, Feb 08 2013

nonn

Subsequence of A075896 (primes of the form 2^n - n^2).
Primes of the form 2^p - p^2 (p = prime) for p = 5, 7, 17, 19, 53, 83, 227, 461, 2221,... (all p <= 2455).
a(8) = 59542628... has 139 digits.
a(9) = 2^2221 - 2221^2 has 669 digits.

			130783 is in the sequence because it is prime and it is of the form 2^p - p^2 where p=17 is prime.

Cf. A075896.

lst={}; Do[p=Prime[n]; If[PrimeQ[p=2^p-p^2], AppendTo[lst, p]], {n, 100}]; lst

cp's OEIS Frontend

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

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A072180 Numbers k such that 2^k - k^2 is prime.

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A061119 Primes in the sequence n^2 + 2^n (A001580).

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A099482 Semiprimes of the form 2^k - k^2.

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A099481 Numbers k such that 2^k - k^2 is a semiprime.

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A192515 Number of primes in the range [2^n-n^2, 2^n].

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Maple

Mathematica

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A182360 Primes of the form 2^n - n^4.

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A222137 Primes of the form 2^p - p^2, where p is prime.

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