A002253 -id:A002253 - cp's OEIS frontend

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

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

A004119 a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.

Original entry on oeis.org

1, 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, 6442450945, 12884901889

Offset: 0

N. J. A. Sloane, R. K. Guy

Also Pisot sequence L(4,7) (cf. A008776).
Alternatively, define the sequence S(a(1),a(2)) by: a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n > 0. This is S(4,7).
a(n) = number of terms of the arithmetic progression with first term 2^(2n-1) and last term 2^(2n+1). So common difference is 2^n. E.g., a(2)=7 corresponds to (8,12,16,20,24,28,32). - Augustine O. Munagi, Feb 21 2007
Equals binomial transform of [1, 3, 0, 3, 0, 3, 0, 3, ...]. - Gary W. Adamson, Aug 27 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
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]
A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Sequences, 11 (2008), #08.5.4.
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
Index entries for linear recurrences with constant coefficients, signature (3, -2).
Index entries for Pisot sequences

Cf. A049988, A049775, A000051, A008776.
A181565 is an essentially identical sequence.
For primes see A002253 and A039687.

[1] cat [n le 1 select 4 else 2*Self(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015

A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation

s=4;lst={1,s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *)
Prepend[Table[3*2^n + 1, {n, 0, 32}], 1] (* or *)
{1}~Join~LinearRecurrence[{3, -2}, {4, 7}, 33] (* Michael De Vlieger, Dec 16 2015 *)

a(n)=3<Charles R Greathouse IV, Sep 28 2015

a(n) = 3a(n-1) - 2a(n-2).
For n>3, a(3)=13, a(n)= a(n-1)+2*floor(a(n-1)/2). - Benoit Cloitre, Aug 14 2002
For n>=1, a(n) = A049775(n+1)/2^(n-2). For n>=2, a(n) = 2a(n-1)-1; see also A000051. - Philippe Deléham, Feb 20 2004
O.g.f.: -(-1-x+3*x^2)/((2*x-1)*(x-1)). - R. J. Mathar, Nov 23 2007
For n>0, a(n) = 2*a(n-1)-1. - Vincenzo Librandi, Dec 16 2015
E.g.f.: exp(x)*(1 + 3*sinh(x)). - Stefano Spezia, May 06 2023

Edited by N. J. A. Sloane, Dec 16 2015 at the suggestion of Bruno Berselli

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

A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

Original entry on oeis.org

7, 11, 13, 19, 37, 41, 67, 73, 97, 131, 137, 193, 521, 577, 641, 769, 1033, 1153, 2053, 2081, 2113, 4099, 4129, 8209, 12289, 16417, 18433, 32771, 32801, 32833, 40961, 65539, 133121, 147457, 163841, 262147, 262153, 262657, 270337, 524353, 524801

Offset: 1

Reinhard Zumkeller, Mar 05 2003

This is sequence A070739 without the Fermat primes, A000215. Sequence A081504 lists the i for which there are no primes. - T. D. Noe, Jun 22 2007

Primes in A014311. - Reinhard Zumkeller, May 03 2012

			    7 = 2^2 + 2^1 + 1
   11 = 2^3 + 2^1 + 1
   13 = 2^3 + 2^2 + 1
   19 = 2^4 + 2^1 + 1
   37 = 2^5 + 2^2 + 1
   41 = 2^5 + 2^3 + 1
   67 = 2^6 + 2^1 + 1
   73 = 2^6 + 2^3 + 1
   97 = 2^6 + 2^5 + 1
  131 = 2^7 + 2^1 + 1
  137 = 2^7 + 2^3 + 1
  193 = 2^7 + 2^6 + 1
  521 = 2^9 + 2^3 + 1

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..7800 (n = 1..1000 from T. D. Noe)
Richard Ehrenborg and N. Bradley Fox, The Descent Set Polynomial Revisited, arXiv:1408.6858 [math.CO], 2014.
Norman B. Fox, Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians, University of Kentucky, Theses and Dissertations, Mathematics, Paper 25.

Essentially the same as A070739.
Cf. A000040, A000215, A081092, A010051.
Cf. A095077 (primes with four bits set).
A057733 = 2^A057732 + 3 and A039687 = 3*2^A002253 + 1 are subsequences.

a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
-- Reinhard Zumkeller, May 03 2012

N:= 20: # to get all terms < 2^N
select(isprime, [seq(seq(2^i+2^j+1,j=1..i-1),i=1..N-1)]); # Robert Israel, May 17 2016

Select[Flatten[Table[2^i + 2^j + 1, {i, 21}, {j, i-1}]], PrimeQ] (* Alonso del Arte, Jan 11 2011 *)

do(mx)=my(v=List(),t); for(i=2,mx,for(j=1,i-1,if(ispseudoprime(t=2^i+2^j+1), listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Jan 02 2014

is(n)=hammingweight(n)==3 && isprime(n) \\ Charles R Greathouse IV, Aug 28 2017

A81091=[7]; next_A081091(p, i=exponent(p), j=exponent(p-2^i))=!until(isprime(2^i+2^j+1), j++>=i && i++ && j=1)+2^i+2^j
A081091(n)={for(k=#A81091, n-1, A81091=concat(A81091, next_A081091(A81091[k]))); A81091[n]} \\ M. F. Hasler, Mar 03 2023

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A081091_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(1) for s in multiset_permutations('0'*(l-1)+'1'))))
A081091_list = list(islice(A081091_gen(),30)) # Chai Wah Wu, Jul 19 2022

A000120(a(n)) = 3.

A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

41, 209, 157169, 213321, 303093, 382449, 2145353, 2478785

Offset: 1

Arkadiusz Wesolowski, Jan 17 2012

Terms are odd: by Morehead's theorem, 3*2^(2*n) + 1 can never divide a Fermat number.
No other terms below 7516000.
Is this sequence the same as "Numbers k such that 3*2^k + 1 is a factor of a Fermat number 2^(2^m) + 1 for some m"? - Arkadiusz Wesolowski, Nov 13 2018
The last sentence of Morehead's paper is: "It is easy to show that composite numbers of the forms 2^kappa * 3 + 1, 2^kappa * 5 + 1 can not be factors of Fermat's numbers." [a proof is needed]. - Jeppe Stig Nielsen, Jul 23 2019
Any factor of a Fermat number 2^(2^m) + 1 of the form 3*2^k + 1 is prime if k < 2*m + 6. - Arkadiusz Wesolowski, Jun 12 2021
If, for any m >= 0, F(m) = 2^(2^m) + 1 has a prime factor p of the form 3*2^k + 1, then F(m)/p is congruent to 11 mod 30. - Arkadiusz Wesolowski, Jun 13 2021
A number k belongs to this sequence if and only if the order of 2 modulo p is not divisible by 3, where p is a prime of the form 3*2^k + 1 (see Golomb paper). - Arkadiusz Wesolowski, Jun 14 2021

Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
Wilfrid Keller, Fermat factoring status
J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002253.

Cf. A000215, A039687, A057775, A057778, A201364, A226366.

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 209, 2}]; lst

isok(n) = my(p = 3*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ Michel Marcus, Nov 10 2018

A002254 Numbers k such that 5*2^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 4687, 5947, 13165, 23473, 26607, 125413, 209787, 240937, 819739, 1282755, 1320487, 1777515

Offset: 1

N. J. A. Sloane

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).

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
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
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.
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.]
Eric Weisstein's World of Mathematics, Proth Prime
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
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050526.

lst={};Do[If[PrimeQ[5*2^n+1], AppendTo[lst, n]], {n, 0, 2*10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

is(n)=isprime(5*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

Corrected (removed incorrect term 40937) and added more terms (from http://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html), Joerg Arndt, Apr 07 2013

A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).

Original entry on oeis.org

1, 2, 6, 18

Offset: 1

M. F. Hasler, Oct 30 2010

Sequences A181491 and A181492 list the corresponding primes.
No more terms below three million. - Charles R Greathouse IV, Mar 14 2011
Intersection of A002235 and A002253. - Jeppe Stig Nielsen, Mar 05 2018

Cf. A001097, A001359, A002235, A002253, A006512, A014574, A294730.

Filtered([1..300],k->IsPrime(3*2^k-1) and IsPrime(3*2^k+1)); # Muniru A Asiru, Mar 11 2018

a:=k->`if`(isprime(3*2^k-1) and isprime(3*2^k+1),k,NULL); seq(a(k),k=1..1000); # Muniru A Asiru, Mar 11 2018

fQ[n_] := PrimeQ[3*2^n - 1] && PrimeQ[3*2^n + 1]; k = 1; lst= {}; While[k < 15001, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ] (* Robert G. Wilson v, Nov 05 2010 *)
Select[Range[20],AllTrue[3*2^#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 24 2014 *)

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

Equals { k | A007283(k) in A014574 } = { k | A153893(k) in A001359 }.

Pari program repaired by Charles R Greathouse IV, Mar 14 2011

A268657 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

Original entry on oeis.org

6, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 408, 438, 534, 2208, 3168, 3189, 3912, 34350, 42294, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 2478785, 5082306, 7033641, 10829346

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

nonn

Wilfrid Keller, private communication, 2008.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..41
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=3)
OEIS Wiki, Generalized Fermat numbers

Cf. A059919, A268658, A204620, A268659, A268660, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

for(k=1,+oo,p=3*2^k+1;if(ispseudoprime(p),t=znorder(Mod(3,p));bitand(t,t-1)==0&&print1(k,", "))) \\ Jeppe Stig Nielsen, Oct 30 2020

A268658 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

Original entry on oeis.org

2, 8, 18, 66, 189, 209, 408, 2208, 2816, 3168, 3912, 20909, 54792, 59973, 157169, 303093, 709968, 801978, 1832496, 2145353, 2291610, 5082306, 10829346, 16408818

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=5)
OEIS Wiki, Generalized Fermat numbers

Cf. A199591, A268657, A204620, A268659, A268660, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

for(k=1,+oo,p=3*2^k+1;if(ispseudoprime(p),t=znorder(Mod(5,p));bitand(t,t-1)==0&&print1(k,", "))) \\ Jeppe Stig Nielsen, Oct 30 2020

a(24) from Jeppe Stig Nielsen, Oct 30 2020

A268660 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Original entry on oeis.org

2, 5, 8, 41, 209, 353, 2816, 20909, 42665, 157169, 213321, 303093, 362765, 382449, 2145353, 2478785

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
OEIS Wiki, Generalized Fermat numbers

Cf. A152585, A268657, A268658, A204620, A268659, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

cp's OEIS Frontend

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

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A004119 a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.

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

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A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

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A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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A002254 Numbers k such that 5*2^k + 1 is prime.

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A181490 Numbers k such that 3*2^k-1 and 3*2^k+1 are twin primes (A001097).

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A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).