A039687 -id:A039687 - cp's OEIS frontend

A259865 Exponent of 2 modulo the prime A039687(n).

3, 12, 48, 96, 384, 6144, 393216, 805306368, 103079215104, 549755813888, 110680464442257309696, 1176956575385002643219210516851437453019191645837006471168

Offset: 1

N. J. A. Sloane, Jul 08 2015

Here we corrected a(10) which was 1649267441664 in Golomb's article. - R. J. Mathar, Jul 14 2015

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [DOI]
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

Cf. A039687.

A259865 := proc(n)
    numtheory[order](2,A039687(n)) ;
end proc: # R. J. Mathar, Jul 14 2015

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

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

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

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

A007505 Primes of form 3*2^n - 1.

Original entry on oeis.org

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407, 59421121885698253195157962751, 30423614405477505635920876929023

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

a(1) = 2, define f(k) = 2k+1, then a(n+1) = least prime fff...(a(n)). After 383 the next terem is 6143. We have f(383) = 767 (composite), f(767) = 1535 (composite), f(1565)=3071(composite), f(3071) = 6143 (prime), hence the next term is 6143= ffff(383). - Amarnath Murthy, Jul 13 2005

If n is in the sequence and m=(n+1)/3 then m is a solution of the equation, sigma(x+sigma(x))=3x (*). Is it true that there is no other solution of (*)? - Farideh Firoozbakht, Dec 05 2005

H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, pp. 381-384.
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 = 1..27
Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Wilfrid Keller, List of primes k*2^n - 1 for k < 300
Amelia Carolina Sparavigna, A recursive formula for Thabit 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.
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Subsequence of A083329.
See A002235 for more terms.
Cf. A039687 (primes of the form 3*2^n+1).
Cf. A010051.

a007505 n = a007505_list !! (n-1)
a007505_list = filter ((== 1) . a010051') a083329_list
-- Reinhard Zumkeller, Sep 10 2013

[a: n in [0..200] | IsPrime(a) where a is 3*2^n-1]; // Vincenzo Librandi, Mar 20 2013

Reap[For[n = 0, n <= 103, n++, If[PrimeQ[p = 3*2^n - 1], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2012 *)
Select[Table[3 2^n - 1, {n, 0, 100}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

for(n=0,100, if(isprime(t=3<Charles R Greathouse IV, Feb 07 2017

a(n) = 3*2^A002235(n)-1. - Zak Seidov, Jul 21 2016

A334092 Primes p of the form of the form q*2^h + 1, where q is one of the Fermat primes; Primes p for which A329697(p) == 2.

Original entry on oeis.org

7, 11, 13, 41, 97, 137, 193, 641, 769, 12289, 40961, 163841, 557057, 786433, 167772161, 2281701377, 3221225473, 206158430209, 2748779069441, 6597069766657, 38280596832649217, 180143985094819841, 221360928884514619393, 188894659314785808547841, 193428131138340667952988161

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p such that p-1 is not a power of two, but for which A171462(p-1) = (p-1-A052126(p-1)) is [a power of 2].

Primes of the form ((2^(2^k))+1)*2^h + 1, where ((2^(2^k))+1) is one of the Fermat primes, A019434, 3, 5, 17, 257, ..., .

Charles R Greathouse IV, Table of n, a(n) for n = 1..53

Primes in A334102.
Intersection of A081091 and A147545.
Subsequences: A039687, A050526, A300407.
Cf. A000040, A010051, A019434, A052126, A171462, A329697. Cf. also A334093, A334094, A334095, A334096.

isA334092(n) = (isprime(n)&&2==A329697(n));

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A209229(n) = (n && !bitand(n,n-1));
isA334092(n) = (isprime(n)&&(!A209229(n-1))&&A209229(n-1-A052126(n-1)));

list(lim)=if(exponent(lim\=1)>=2^33, error("Verify composite character of more Fermat primes before checking this high")); my(v=List(),t); for(e=0,4, t=2^2^e+1; while((t<<=1)Charles R Greathouse IV, Apr 14 2020

More terms from Giovanni Resta, Apr 14 2020

A074781 Primes of the form p*2^k + 1 for any k and any prime p.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 149, 167, 173, 179, 193, 227, 233, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 449, 467, 479, 503, 509, 557, 563, 569, 587, 593, 641, 653, 719, 769, 773, 797, 809, 839, 857, 863, 887

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

nonn

From Bernard Schott, Dec 14 2020: (Start)
Equivalently, primes p such that the ratio (p-1)/gpf(p-1) = 2^k where gpf(m) is the greatest prime factor of m, A006530.
Paul Erdős asked if there are infinitely many primes p in this sequence (see R. K. Guy reference). (End)

			3 = 2*2^0+1 is a term and 2/2 = 1 = 2^0.
7 = 3*2^1+1 is a term and 6/3 = 2 = 2^1.
13 = 3*2^2+1 is a term and 12/3 = 4 = 2^2.
41 = 5*2^3+1 is a term and 40/5 = 8 = 2^3.
113 = 7*2^4+1 is a term and 112/7 = 16 = 2^4.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

T. D. Noe, Table of n, a(n) for n = 1..2000
Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8.

Cf. A000079, A006093, A006530, A052126, A339464.
Cf. other ratios : A339463, A339465, A339466.
Subsequences: A039687, A051900, A058500 (this sequence without the Fermat primes), A090866, A147545,

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:
3, op(select(is_a, [$3..919])); # Peter Luschny, Dec 14 2020

Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)

A050526 Primes of form 5*2^n+1.

Original entry on oeis.org

11, 41, 641, 40961, 163841, 167772161, 2748779069441, 180143985094819841, 188894659314785808547841, 193428131138340667952988161, 850705917302346158658436518579420528641

Offset: 1

N. J. A. Sloane, Dec 29 1999

nonn

All terms are odd since if n is even, then 5*2^n+1 is divisible by 3. - Michele Fabbrini, Jun 06 2021

Muniru A Asiru, Table of n, a(n) for n = 1..12
Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300
Wikipedia, Proth's theorem
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

For the corresponding exponents n see A002254.

Cf. A019434, A039687, A050527, A050528, A050529, A195745, A300406, A300407, A300408.

Filtered(List([1..270], n->5*2^n + 1), IsPrime); # Muniru A Asiru, Mar 06 2018

[a: n in [1..200] | IsPrime(a) where a is 5*2^n + 1]; // Vincenzo Librandi, Mar 06 2018

a:=(n, k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
seq(a(n, 5), n=1..127); # Martin Renner, Mar 05 2018

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=5*2^k+1), print1(p, ", "))); } \\ Altug Alkan, Mar 29 2018

a(n) = A083575(A002254(n)). - Michel Marcus, Mar 29 2018

cp's OEIS Frontend

A259865 Exponent of 2 modulo the prime A039687(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A007505 Primes of form 3*2^n - 1.

Views

Author