A163115 -id:A163115 - cp's OEIS frontend

A176691 a(n) = 2^n + 2*n + 1.

2, 5, 9, 15, 25, 43, 77, 143, 273, 531, 1045, 2071, 4121, 8219, 16413, 32799, 65569, 131107, 262181, 524327, 1048617, 2097195, 4194349, 8388655, 16777265, 33554483, 67108917, 134217783, 268435513, 536870971, 1073741885, 2147483711, 4294967361, 8589934659, 17179869253

Offset: 0

Jonathan Vos Post, Apr 23 2010

The subsequence of primes in this sequence is A163115.

Also the number of connected dominating sets in the (n+1)-wheel graph. - Eric W. Weisstein, Aug 30 2017

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Wheel Graph
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000051, A000079, A000290, A005126, A005843, A163115, A194455.

List([0..35],n->2^n+2*n+1); # Muniru A Asiru, Mar 25 2018

[2^n + 2*n + 1: n in [0..40]]; // Vincenzo Librandi, Aug 12 2015

seq(2^n+2*n+1,n=0..35); # Muniru A Asiru, Mar 25 2018

Table[2^n + 2 n + 1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
LinearRecurrence[{4, -5, 2}, {2, 5, 9}, 40] (* Vincenzo Librandi, Aug 12 2015 *)
CoefficientList[Series[(-2 + 3 x + x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 30 2017 *)

vector(40, n, n--; 2^n + 2*n + 1) \\ Michel Marcus, Aug 12 2015

a(n) = 2^n + 2*n + 1 = A000079(n) + A005843(n) + 1 = A000051(n) + A005843(n).
From R. J. Mathar, Apr 28 2010: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (-2 + 3*x + x^2)/((2*x - 1)*(x - 1)^2). (End)
E.g.f.: exp(x)*(1 + exp(x) + 2*x). - Stefano Spezia, May 06 2023

Corrected (one 1048617 replaced by 2097195) by R. J. Mathar, Apr 28 2010

A192436 Primes of the form 2^n+2*n-3.

Original entry on oeis.org

5, 11, 73, 139, 269, 1048613, 67108913, 134217779, 4294967357, 549755813963, 9223372036854775931, 2417851639229258349412511, 316912650057057350374175801537, 2596148429267413814265248164610267

Offset: 1

Juri-Stepan Gerasimov, Jul 03 2011

nonn

Generated by n = 2, 3, 6, 7, 8, 20, 26, 27, 32, 39, 63, 81, 98, 111,...

Cf. A163115.

A301634 Numbers k such that 2^k + 2*k + 1 is prime.

Original entry on oeis.org

0, 1, 5, 13, 65, 85, 229, 2005, 3875, 3919, 5417, 8819, 11899, 16668, 19445, 28242, 33407, 37918, 40594, 141251

Offset: 1

Seiichi Manyama, Mar 25 2018

Next term, if it exists, is greater than 50000. Terms up to 229 correspond to provable primes. The terms greater than or equal to 2005 correspond to probable primes. - Jon E. Schoenfield and Vaclav Kotesovec, Mar 27 2018

A163115 gives the primes.
Numbers k such that b^k + b*k + 1 is prime: this sequence (b=2), A171058 (b=3), A301635 (b=5).
Cf. A176691.

a:=k->`if`(isprime(2^k+2*k+1),k,NULL): seq(a(k),k=0..6000); # Muniru A Asiru, Mar 25 2018

Flatten[{0, Select[Range[5000], PrimeQ[2^# + 2*# + 1] &]}] (* Vaclav Kotesovec, Mar 25 2018 *)

for(n=0, 500, if(isprime(2^n+2*n+1), print1(n", ")))

a(9)-a(15) from Vaclav Kotesovec, Mar 25 2018
a(16), a(18)-a(19) from Jon E. Schoenfield, Mar 26 2018
a(17) inserted by and a(20) from Michael S. Branicky, Jun 23 2024

A192764 Numbers k such that 2^(k-1)+2*k-1 is a prime number.

Original entry on oeis.org

1, 2, 6, 14, 66, 86, 230, 2006, 3876, 3920, 5418, 8820, 11900, 16669, 19446, 28243, 33408, 37919, 40595

Offset: 1

Juri-Stepan Gerasimov, Jul 09 2011

a(20) > 10^5 if it exists. - Michael S. Branicky, Aug 26 2024

Cf. A004051, A163115, A176691, A192436.

Select[Range[4000], PrimeQ[2^(# - 1) + 2# - 1] &] (* Alonso del Arte, Jul 09 2011 *)

for(n=1,10^6,if(ispseudoprime(2^(n-1)+2*n-1),print1(n,", ")));

a(13)-a(19) from Michael S. Branicky, Jul 14 2023

A301638 Primes of the form 5^k + 5*k + 1.

Original entry on oeis.org

2, 11, 48828181, 298023223876953251, 2910383045673370361328301, 258493941422821148397315216271863391739316284656524658203551

Offset: 1

Seiichi Manyama, Mar 25 2018

nonn

The next term is too large to include.

The next term has 349 digits. - Harvey P. Dale, Dec 03 2018

Primes of the form b^k+b*k+1: A163115 (b=2), A180269 (b=3), this sequence (b=5).

Cf. A301635.

Select[Table[5^n+5n+1,{n,0,550}],PrimeQ] (* Harvey P. Dale, Dec 03 2018 *)

a(n) = 5^A301635(n) + 5*A301635(n) + 1.

A280846 Numbers k such that all four of the numbers 2^k +- 2k +- 1 are nonprime.

Original entry on oeis.org

9, 10, 16, 17, 19, 20, 21, 22, 23, 24, 26, 29, 30, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 86, 87, 88, 89, 90, 91

Offset: 1

Maverick K. Morrison, Jan 15 2017

nonn

			For k=9, 2^k +- 2k +- 1 produces 531, 529, 495, and 493, none of which are prime.

Cf. A061761 (numbers of the form 2^n + 2*n - 1), A105330 (numbers n such that 2^(n+1) + 2n + 1 is prime), A163115 (primes of the form 2^n + 2*n + 1), A173168 (primes of the form 2^k + 2k - 1). - Jon E. Schoenfield, Jan 22 2017

Select[Range[100],NoneTrue[Flatten[{2^#+2#+{1,-1},2^#-2#+{1,-1}}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2018 *)

cp's OEIS Frontend

A176691 a(n) = 2^n + 2*n + 1.

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A192436 Primes of the form 2^n+2*n-3.

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A301634 Numbers k such that 2^k + 2*k + 1 is prime.

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A192764 Numbers k such that 2^(k-1)+2*k-1 is a prime number.

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A301638 Primes of the form 5^k + 5*k + 1.

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A280846 Numbers k such that all four of the numbers 2^k +- 2k +- 1 are nonprime.

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