A099439 -id:A099439 - cp's OEIS frontend

A099440 Primes of the form A000295(k) = 2^k - k - 1.

11, 1013, 16369, 65519, 67108837, 1125899906842573, 72057594037927879, 1180591620717411303353, 83076749736557242056487941267521419

Offset: 1

Hugo Pfoertner, Oct 18 2004

nonn

The next term a(10) = 2^2072-2073 has 624 decimal digits.

a(11) has 1882 decimal digits. - Vincenzo Librandi, Jul 18 2012

			a(2) = 1013 because A000295(A099439(2)) = 2^10 - 10 - 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..10

Cf. A000295 2^n-n-1 (column 2 of the Eulerian numbers), A099439 2^n-n-1 is prime, A099441 2^n-n-1 is a semiprime, A099442 semiprimes in A000295.

[ a: n in [1..200] | IsPrime(a) where a is 2^n-n-1 ]; // Vincenzo Librandi, Jul 18 2012

Select[Table[2^n-n-1,{n,0,7000}],PrimeQ] (* Vincenzo Librandi, Jul 18 2012 *)

A099442 Semiprimes of the form 2^k-k-1.

Original entry on oeis.org

4, 26, 57, 247, 502, 4083, 1073741793, 4294967263, 8589934558, 70368744177617, 4503599627370443, 4611686018427387841, 18889465931478580854709, 75557863725914323419059, 77371252455336267181195177, 316912650057057350374175801245

Offset: 1

Hugo Pfoertner, Oct 18 2004

nonn

Semiprimes in A000295.

			a(4) = 247 because 247 = 13*19 = 2^8-8-1 = 2^A099441(4)-A099441(4)-1.

Hugo Pfoertner, Table of n, a(n) for n = 1..31 (first 19 terms from Vincenzo Librandi)

Cf. A099439, A099440, A099441.

Select[Table[2^n - n - 1, {n, 300}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)

from sympy.ntheory.factor_ import primeomega
def ok(n): return primeomega(2**n-n-1) == 2
print([2**m-m-1 for m in range(2, 100) if ok(m)]) # Michael S. Branicky, Apr 26 2021

A099441 Numbers n such that A000295(n) = 2^n-n-1 is a semiprime.

Original entry on oeis.org

3, 5, 6, 8, 9, 12, 30, 32, 33, 46, 52, 62, 74, 76, 86, 98, 130, 134, 154, 228, 230, 242, 256, 266, 346, 352, 382, 412, 428, 474, 488, 634, 650, 662, 688, 704, 722, 772, 896, 986, 1108, 1222, 1246, 1326

Offset: 1

Hugo Pfoertner, Oct 18 2004

A candidate for the next term is a(45) = 1736. 2^1736-1737 is composite with 523 decimal digits and unknown factorization. - Tyler Busby, Jan 05 2025

			a(3) = 6 because 2^6-6-1 = 57 = 3*19 is a semiprime.

Dario Alpern, Factorization using the Elliptic Curve Method.
Status of 2^1736-1737 in factordb.com.

Cf. A000295 (2^n-n-1 (column 2 of the Eulerian numbers)), A099439 (2^n-n-1 is prime), A099440 (primes in A000295), A099442 (semiprimes in A000295).

More terms from Hugo Pfoertner, Aug 13 2007
a(24) corrected by Hugo Pfoertner, Sep 07 2017
a(32)-a(44) from Tyler Busby, Jan 05 2025

A063791 Numbers k such that 2^(k+1) - k - 2 is prime.

Original entry on oeis.org

3, 9, 13, 15, 25, 49, 55, 69, 115, 2071, 6249, 13669, 14215, 14625, 396127

Offset: 1

Jason Earls, Aug 17 2001

Dean Hickerson, personal communication.

Cf. A099439.

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

a(n) = A099439(n) - 1. - Hugo Pfoertner, Jul 24 2019

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

a(15) from Karsten Bonath, Jun 07 2018

A296031 Numbers k such that 2^(k-1) - k is prime.

Original entry on oeis.org

5, 11, 15, 17, 27, 51, 57, 71, 117, 2073, 6251, 13671, 14217, 14627, 396128

Offset: 1

Thomas Gajdek, Dec 03 2017

a(15) > 200000. - Giovanni Resta, May 13 2018

			5 is in the sequence, because 2^4 - 5 = 11 is prime.

Cf. A063791, A099439.

Select[Range[6500], PrimeQ[2^(# - 1) - #] &] (* Michael De Vlieger, Apr 21 2018 *)

forstep(n=1, 10^6, 2, if(ispseudoprime(2^(n-1)-n),print1(n,", "))); \\ Joerg Arndt, Apr 15 2018

a(n) = A099439(n) + 1 = A063791(n) + 2.

Edited by Joerg Arndt, Apr 15 2018

a(15) from Michael S. Branicky, Apr 20 2025 using A099439

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A099440 Primes of the form A000295(k) = 2^k - k - 1.

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A099442 Semiprimes of the form 2^k-k-1.

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A099441 Numbers n such that A000295(n) = 2^n-n-1 is a semiprime.

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A063791 Numbers k such that 2^(k+1) - k - 2 is prime.

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A296031 Numbers k such that 2^(k-1) - k is prime.

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