A004119 -id:A004119 - cp's OEIS frontend

A000051 a(n) = 2^n + 1.

2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593

Offset: 0

N. J. A. Sloane

Same as Pisot sequence L(2,3).
Length of the continued fraction for Sum_{k=0..n} 1/3^(2^k). - Benoit Cloitre, Nov 12 2003
See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - Philippe Deléham, Feb 20 2004
From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg, May 31 2005
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)* charpoly(A,3). - Milan Janjic, Jan 27 2010
First differences of A006127. - Reinhard Zumkeller, Apr 14 2011
The odd prime numbers in this sequence form A019434, the Fermat primes. - David W. Wilson, Nov 16 2011
Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012
Is the mentioned Pisano period lengths (see above) the same as A007733? - Omar E. Pol, Aug 10 2012
Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012
For n >= 1, a(n) is the total length of the segments of the Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014
Frénicle de Bessy (1657) proved that a(3) = 9 is the only square in this sequence. - Charles R Greathouse IV, May 13 2014
a(n) is the number of distinct possible sums made with at most two elements in {1,...,a(n-1)} for n > 0. - Derek Orr, Dec 13 2014
For n > 0, given any set of a(n) lattice points in R^n, there exist 2 distinct members in this set whose midpoint is also a lattice point. - Melvin Peralta, Jan 28 2017
Also the number of independent vertex sets, irredundant sets, and vertex covers in the (n+1)-star graph. - Eric W. Weisstein, Aug 04 and Sep 21 2017
Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - Eric W. Weisstein, Dec 31 2017
Conjecture: For any integer n >= 0, a(n) is the permanent of the (n+1) X (n+1) matrix with M(j, k) = -floor((j - k - 1)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 46, 60, 244.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.

Ivan Panchenko, Table of n, a(n) for n = 0..100
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Bartomeu Fiol, Jairo Martínez-Montoya, and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
Kival Ngaokrajang, Illustration of Hilbert curve for n = 1..5
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 1.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Crossed Prism Graph.
Eric Weisstein's World of Mathematics, Cunningham Number.
Eric Weisstein's World of Mathematics, Fermat-Lucas Number.
Eric Weisstein's World of Mathematics, Hilbert curve.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Irredundant Set.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.
Eric Weisstein's World of Mathematics, Star Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from the initial 1, identical to A094373.
See A008776 for definitions of Pisot sequences.
Column 2 of array A103438.
Cf. A007583 (a((n-1)/2)/3 for odd n).
Cf. A000079, A000225, A000918, A004119, A005126, A006217, A006521, A007583.
Cf. A007689, A007733, A019434, A024036, A027641, A027642, A034472, A034474.
Cf. A034491, A034524, A036968, A052539, A052944, A062394, A062395, A062396.
Cf. A062397, A063376, A063481, A074600 - A074624, A127904, A173786, A176691.
Cf. A178248, A194455, A228081, A323482.

a000051 = (+ 1) . a000079
a000051_list = iterate ((subtract 1) . (* 2)) 2
-- Reinhard Zumkeller, May 03 2012

[2^n+1: n in [0..40]]; // G. C. Greubel, Jan 18 2025

A000051:=-(-2+3*z)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation
a := n -> add(binomial(n,k)*bernoulli(n-k,1)*2^(k+1)/(k+1),k=0..n); # Peter Luschny, Apr 20 2009

Table[2^n + 1, {n,0,40}]
2^Range[0,40] + 1 (* Eric W. Weisstein, Jul 17 2017 *)
LinearRecurrence[{3, -2}, {2, 3}, 40] (* Eric W. Weisstein, Sep 21 2017 *)

PARI
```
a(n)=2^n+1
```

first(n) = Vec((2 - 3*x)/((1 - x)*(1 - 2*x)) + O(x^n)) \\ Iain Fox, Dec 31 2017

def A000051(n): return (1<Chai Wah Wu, Dec 21 2022

a(n) = 2*a(n-1) - 1 = 3*a(n-1) - 2*a(n-2).
G.f.: (2-3*x)/((1-x)*(1-2*x)).
First differences of A052944. - Emeric Deutsch, Mar 04 2004
a(0) = 1, then a(n) = (Sum_{i=0..n-1} a(i)) - (n-2). - Gerald McGarvey, Jul 10 2004
Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - Ross La Haye, Feb 07 2005
a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007
Equals binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = A000079(n)+1. - Omar E. Pol, May 18 2008
E.g.f.: exp(x) + exp(2*x). - Mohammad K. Azarian, Jan 02 2009
a(n) = A024036(n)/A000225(n). - Reinhard Zumkeller, Feb 14 2009
From Peter Luschny, Apr 20 2009: (Start)
A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)
a(n) is the a(n-1)-th odd number for n >= 1. - Jaroslav Krizek, Apr 25 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n)*A000225(n) = A000225(2*n).
a(n) = A173786(n,0). (End)
If p[i]=Fibonacci(i-4) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise, then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n+2) = a(n) + a(n+1) + A000225(n). - Ivan N. Ianakiev, Jun 24 2012
a(A006521(n)) mod A006521(n) = 0. - Reinhard Zumkeller, Jul 17 2014
a(n) = 3*A007583((n-1)/2) for n odd. - Eric W. Weisstein, Jul 17 2017
Sum_{n>=0} 1/a(n) = A323482. - Amiram Eldar, Nov 11 2020

A181565 a(n) = 3*2^n + 1.

Original entry on oeis.org

4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473

Offset: 0

M. F. Hasler, Oct 30 2010

From Peter Bala, Oct 28 2013: (Start)
Let x and b be positive real numbers. We define an Engel expansion of x to the base b to be a (possibly infinite) nondecreasing sequence of positive integers [a(1), a(2), a(3), ...] such that we have the series representation x = b/a(1) + b^2/(a(1)*a(2)) + b^3/(a(1)*a(2)*a(3)) + .... Depending on the values of x and b such an expansion may not exist, and if it does exist it may not be unique. When b = 1 we recover the ordinary Engel expansion of x.
This sequence gives an Engel expansion of 2/3 to the base 2, with the associated series expansion 2/3 = 2/4 + 2^2/(4*7) + 2^3/(4*7*13) + 2^4/(4*7*13*25) + ....
More generally, for n and m positive integers, the sequence [m + 1, n*m + 1, n^2*m + 1, ...] gives an Engel expansion of the rational number n/m to the base n. See the cross references for several examples. (End)
The only squares in this sequence are 4, 25, 49. - Antti Karttunen, Sep 24 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Hunar Sherzad Taher and Saroj Kumar Dash, On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 448-459. See p. 2.
Wikipedia, Engel Expansion
Index entries for linear recurrences with constant coefficients, signature (3,-2)

Cf. A007283, A020737, A083575, A083683, A083686, A168596, A083705, A195744.
Essentially a duplicate of A004119.
A002253 and A039687 give the primes in this sequence, and A181492 is the subsequence of twin primes.

[3*2^n + 1: n in [0..30]]; // Vincenzo Librandi, May 19 2011

3*2^Range[0,50]+1 (* Vladimir Joseph Stephan Orlovsky, Mar 24 2011 *)
LinearRecurrence[{3,-2},{4,7},40] (* Harvey P. Dale, Sep 19 2024 *)

PARI
```
A181565(n)=3<
				
```

a(n) = A004119(n+1) = A103204(n+1) for all n >= 0.
From Ilya Gutkovskiy, Jun 01 2016: (Start)
O.g.f.: (4 - 5*x)/((1 - x)*(1 - 2*x)).
E.g.f.: (1 + 3*exp(x))*exp(x).
a(n) = 3*a(n-1) - 2*a(n-2). (End)
a(n) = 2*a(n-1) - 1. - Miquel Cerda, Aug 16 2016
For n >= 0, A005940(a(n)) = A001248(1+n). - Antti Karttunen, Sep 24 2023

A062709 a(n) = 2^n + 3.

Original entry on oeis.org

4, 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027, 2051, 4099, 8195, 16387, 32771, 65539, 131075, 262147, 524291, 1048579, 2097155, 4194307, 8388611, 16777219, 33554435, 67108867, 134217731, 268435459, 536870915, 1073741827, 2147483651, 4294967299, 8589934595, 17179869187

Offset: 0

Henry Bottomley, Jul 13 2001

Written in binary a(n) is 1000...00011 for n > 1.

For n >= 2, a(n) is the minimal k for which A000120(k(2^n-1)) is not multiple of n. - Vladimir Shevelev, Jun 05 2009

			a(3) = 2^3 + 3 = 8 + 3 = 11.
a(4) = 2^4 + 3 = 16 + 3 = 19.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Primes in this sequence are A057733.

Cf. A000051, A000079, A000120, A000225, A004119, A030101, A052548, A173921.

[2^n+3: n in [0..40]] // Vincenzo Librandi, Jan 31 2012

LinearRecurrence[{3, -2}, {4, 5}, 40] (* Vincenzo Librandi, Jan 31 2012 *)
NestList[2 * # - 3 &, 4, 20] (* Zak Seidov, Mar 28 2015 *)
2^Range[0, 29] + 3 (* Alonso del Arte, Mar 28 2015 *)

a(n)=2^n+3 \\ Charles R Greathouse IV, Jan 30 2012

[gaussian_binomial(n,1,2)+4 for n in range(0,32)] # Zerinvary Lajos, May 31 2009

a(n) = 2a(n-1) - 3 = A052548(n) + 1 = A000051(n) + 2 = A000079(n) + 3 = A000225(n) + 4 = A030101(A004119(n)) for n > 1.
G.f.: (4 - 7*x)/((1 - 2*x)*(1 - x)).
a(n) = A173921(A000051(n+1)). - Reinhard Zumkeller, Mar 04 2010
E.g.f.: exp(x)*(3 + exp(x)). - Stefano Spezia, May 06 2023

A002253 Numbers k such that 3*2^k + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353

Offset: 1

N. J. A. Sloane

From Zak Seidov, Mar 08 2009: (Start)
List is complete up to 3941000 according to the list of RB & WK.
So far there are only 4 primes: 2, 5, 41, 353. (End)

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.
H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
PrimeGrid, 321 Prime Search statistics
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254, A004119, A181565.

See A039687 for the actual primes.

is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<M. F. Hasler, Mar 03 2023

a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023

Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 2009
Edited by N. J. A. Sloane, Mar 13 2009
a(47) and a(48) from the Ballinger & Keller web page, Joerg Arndt, Apr 07 2013
a(49) from https://t5k.org/primes/page.php?id=116922, Fabrice Le Foulher, Mar 09 2014
Terms moved from Data to b-file (Links), and additional term appended to b-file, by Jeppe Stig Nielsen, Oct 30 2020

A039687 Primes of the form 3*2^k + 1.

Original entry on oeis.org

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, 2353913150770005286438421033702874906038383291674012942337

Offset: 1

Felice Russo

nonn

Primes of the form 6m+1 (A002476) of A074781. - Bernard Schott, Dec 14 2020

S. W. Golomb, Properties of the sequence 3*2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3*2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

For more terms see A002253. These are the primes in A004119 (or A181565).
Cf. A002476, A074781.
Subsequence of A081091.

Select[Table[6 2^n+1,{n,0,250}],PrimeQ] (* Harvey P. Dale, Dec 17 2010 *)

A039687(n)=3<<A002253(n)+1 \\ M. F. Hasler, Mar 03 2023

a(n) = 3*2^A002253(n) + 1. - M. F. Hasler, Mar 03 2023

A049775 a(n) is the sum of all integers from 2^(n-2)+1 to 2^(n-1).

Original entry on oeis.org

2, 7, 26, 100, 392, 1552, 6176, 24640, 98432, 393472, 1573376, 6292480, 25167872, 100667392, 402661376, 1610629120, 6442483712, 25769869312, 103079346176, 412317122560, 1649267965952, 6597070815232, 26388281163776

Offset: 2

Clark Kimberling

nonn

Name when submitted: Sum of even-indexed terms of n-th row of array T given by A049773 (from Clark Kimberling).
Also sum of integers of which the binary order [A029837] is n: a(n) = Sum_[x | ceiling(log_2(x)) = n ]. E.g., a(7) = 6176 = Apply[Plus, Table[w,{w,65,128}]].
This sequence may be obtained by filling a complete binary tree left-to-right, row by row with the integers onwards from 2 and then collecting the sums of the rows; e.g., 2, 3+4, 5+6+7+8, 9+10+11+12+13+14+15+16, etc. a(n) is then equal to the sum of row n-1. - Carl R. White, Aug 19 2003
If the offset is set to zero, the inverse binomial transform gives A007051 without its leading 1. - R. J. Mathar, Mar 26 2009

			a(2) = 2 = 2.
a(3) = 7 = 3 + 4.
a(4) =26 = 5 + 6 + 7 + 8.
..

Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A049773 (sequence motivating the original definition).
Cf. A049775(n+2) = A007582(n+1) - A007582(n).
Cf. A029837, A003070.

LinearRecurrence[{6,-8},{2,7},30] (* Harvey P. Dale, Mar 04 2013 *)

a(n) = 2^(n-3)*(3*2^(n-2)+1). - Carl R. White, Aug 19 2003
From Philippe Deléham, Feb 20 2004: (Start)
a(n+1) = 4*a(n) - 2^(n-2); see also A007582.
a(n+1) = 2^(n-2)*A004119(n). (End)
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 6*a(n-1) - 8*a(n-2).
G.f.: -x^2*(-2+5*x)/((4*x-1)*(2*x-1)). (End)

More terms from Michael Somos

Name change by Olivier Gérard, Oct 24 2017

A056807 Numbers k such that 3*10^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 10, 28, 36, 67, 81, 147, 483, 643, 1020, 1900, 2620, 10453, 27720, 52824, 105589, 111988, 618853, 665829

Offset: 1

Robert G. Wilson v, Aug 22 2000

			k = 3 gives (3*10^3+1) = 3000+1 = 3001, which is prime.

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Makoto Kamada, Prime numbers of the form 300...001.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A056797, A062339, A101823, A199683, A259866.

Do[ If[ PrimeQ[ 3*10^k + 1], Print[ k ]], {k, 0, 20000}]

is(k)=isprime(3*10^k+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A101823(n) + 1.

a(13)-a(14) from Julien Peter Benney (jpbenney(AT)ftml.net), Nov 23 2004
a(15) from Hugo Pfoertner, Jan 18 2005
a(16)-a(17) from Robert G. Wilson v, Jan 18 2005
a(18) from Roman Makarchuk, Dec 05 2008 confirmed as next term by Ray Chandler, Mar 02 2012
a(19) from Alexander Gramolin, Feb 24 2012 confirmed as next term by Ray Chandler, Mar 02 2012
a(20)-a(21) from Kamada data by Robert Price, Jan 26 2015

A181492 Primes of the form p=3*2^k+1 such that p-2 is also a prime.

Original entry on oeis.org

7, 13, 193, 786433

Offset: 1

M. F. Hasler, Oct 30 2010

nonn

Sequence A181490 lists the exponents k, sequences A181491 and A181493 the corresponding lesser twin prime and their average.

a(5) > 3 * 2^3000 + 1. - Max Z. Scialabba, Dec 24 2023

Cf. A001097, A001359, A006512, A014574.

Select[3 2^Range[100]+1,And@@PrimeQ[{#,#-2}]&] (* Harvey P. Dale, Jun 19 2013 *)

PARI
```
for( k=1,999, ispseudoprime(3<
				
```

A181492 = A181491 + 2 = A181493 + 1 = 3*2^A181490 + 1 = intersection of A004119 or A103204 or A181495 with A006512 or A001097.

A204713 T(n,k) is the number of (n+1) X (k+1) 0..1 arrays with the permanents of all 2 X 2 subblocks equal and nonzero.

Original entry on oeis.org

7, 13, 13, 25, 33, 25, 49, 81, 81, 49, 97, 209, 257, 209, 97, 193, 529, 833, 833, 529, 193, 385, 1361, 2689, 3473, 2689, 1361, 385, 769, 3473, 8705, 14145, 14145, 8705, 3473, 769, 1537, 8913, 28161, 58449, 73345, 58449, 28161, 8913, 1537, 3073, 22801, 91137

Offset: 1

R. H. Hardin, Jan 18 2012

This is A183688+1 (the +1 comes from the all-1 matrix). [Discovered by Sequence Machine] - Andrey Zabolotskiy, Oct 19 2021

			Table starts
    7   13    25     49       97       193        385         769         1537
   13   33    81    209      529      1361       3473        8913        22801
   25   81   257    833     2689      8705      28161       91137       294913
   49  209   833   3473    14145     58449     239425      986129      4047681
   97  529  2689  14145    73345    382849    1992321    10382977     54072961
  193 1361  8705  58449   382849   2542369   16748161   110871041    731709057
  385 3473 28161 239425  1992321  16748161  140090241  1174759297   9838208513
  769 8913 91137 986129 10382977 110871041 1174759297 12503757969 132720731393
Some solutions for n=4 k=3
  1  1  1  1    1  1  1  1    0  1  1  1    1  1  1  1    0  1  0  1
  0  1  0  1    0  1  0  1    1  0  1  0    0  1  0  1    1  1  1  0
  1  1  1  0    1  0  1  0    1  1  1  1    1  1  1  0    1  0  1  1
  0  1  0  1    1  1  0  1    0  1  0  1    1  0  1  1    0  1  1  0
  1  1  1  0    0  1  1  0    1  1  1  0    1  1  0  1    1  1  0  1

R. H. Hardin, Table of n, a(n) for n = 1..760
Jon Maiga, Computer-generated formulas for A204713, Sequence Machine.

Column 1 is A004119(n+1).

Cf. A183688.

Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2)
k=2: a(n) = 2*a(n-1) +3*a(n-2) -4*a(n-3)
k=3: a(n) = 3*a(n-1) +2*a(n-2) -4*a(n-3)
k=4: a(n) = a(n-1) +13*a(n-2) +3*a(n-3) -16*a(n-4)
k=5: a(n) = 4*a(n-1) +15*a(n-2) -38*a(n-3) -52*a(n-4) +72*a(n-5)
k=6: (order 9 recurrence)
k=7: (order 10 recurrence)

A103204 a(1) = 2, a(2) = 4; a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

2, 4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473

Offset: 1

Roger L. Bagula, Mar 19 2005

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A003945, A004119.

a[1] = 2; a[2] = 4; a[n_] := a[n] = 2*a[n - 1] - 1; Table[a[n], {n, 1, 32}]
Join[{2},NestList[2#-1&,4,40]] (* or *) LinearRecurrence[{3,-2},{2,4,7},40] (* Harvey P. Dale, Dec 04 2018 *)

Vec(x*(2-2*x-x^2)/(1-3*x+2*x^2) + O(x^50)) \\ Michel Marcus, Jan 29 2016

a(n) = A003945(n-1) + 1.
a(n) = 3*2^(n-2) + 1 for n>1. - Ralf Stephan, May 18 2007
a(n) = A004119(n-1), n>1. - R. J. Mathar, Jun 11 2010
G.f.: x*(2-2*x-x^2)/(1-3*x+2*x^2). a(n) = 3*a(n-1)-2*a(n-2), n>3. - Colin Barker, Jan 29 2012

cp's OEIS Frontend

A000051 a(n) = 2^n + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

A181565 a(n) = 3*2^n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A062709 a(n) = 2^n + 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A002253 Numbers k such that 3*2^k + 1 is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A039687 Primes of the form 3*2^k + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A049775 a(n) is the sum of all integers from 2^(n-2)+1 to 2^(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions