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Search: a039687 -id:a039687
Displaying 1-10 of 19 results found. page 1 2
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A259865 Exponent of 2 modulo the prime A039687(n). +20
1
3, 12, 48, 96, 384, 6144, 393216, 805306368, 103079215104, 549755813888, 110680464442257309696, 1176956575385002643219210516851437453019191645837006471168 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Here we corrected a(10) which was 1649267441664 in Golomb's article. - R. J. Mathar, Jul 14 2015
LINKS
Table of n, a(n) for n=1..12.
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]
MAPLE
A259865 := proc(n)
numtheory[order](2, A039687(n)) ;
end proc: # R. J. Mathar, Jul 14 2015
CROSSREFS
Cf. A039687.
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Jul 08 2015
STATUS
approved
A181565 a(n) = 3*2^n + 1. +10
40
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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]
Wikipedia, Engel Expansion
Index entries for linear recurrences with constant coefficients, signature (3,-2)
FORMULA
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
MATHEMATICA
3*2^Range[0, 50]+1 (* Vladimir Joseph Stephan Orlovsky, Mar 24 2011 *)
PROG
(PARI) A181565(n)=3<<n+1
(Magma) [3*2^n + 1: n in [0..30]]; // Vincenzo Librandi, May 19 2011
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
M. F. Hasler, Oct 30 2010
STATUS
approved
A002253 Numbers k such that 3*2^k + 1 is prime.
(Formerly M1318 N0506) 		+10
22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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)
REFERENCES
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).
LINKS
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
FORMULA
a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023
PROG
(PARI) is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017
(PARI) A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<<m+++1), ); A2253=concat(A2253, m))+A2253[n] \\ M. F. Hasler, Mar 03 2023
CROSSREFS
Cf. A002254, A004119, A181565.
See A039687 for the actual primes.
KEYWORD
hard,nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
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
STATUS
approved
A004119 a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.
(Formerly M3308) 		+10
19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
FORMULA
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
MAPLE
A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation
MATHEMATICA
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 *)
PROG
(PARI) a(n)=3<<n+1 \\ Charles R Greathouse IV, Sep 28 2015
(Magma) [1] cat [n le 1 select 4 else 2*Self(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015
CROSSREFS
Cf. A049988, A049775, A000051, A008776.
A181565 is an essentially identical sequence.
For primes see A002253 and A039687.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, R. K. Guy
EXTENSIONS
Edited by N. J. A. Sloane, Dec 16 2015 at the suggestion of Bruno Berselli
STATUS
approved
A081091 Primes of the form 2^i + 2^j + 1, i > j > 0. +10
19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
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.
FORMULA
A000120(a(n)) = 3.
EXAMPLE
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
MAPLE
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
MATHEMATICA
Select[Flatten[Table[2^i + 2^j + 1, {i, 21}, {j, i-1}]], PrimeQ] (* Alonso del Arte, Jan 11 2011 *)
PROG
(PARI) 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
(PARI) is(n)=hammingweight(n)==3 && isprime(n) \\ Charles R Greathouse IV, Aug 28 2017
(PARI) 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
(Haskell)
a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
-- Reinhard Zumkeller, May 03 2012
(Python)
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
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Mar 05 2003
STATUS
approved
A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m. +10
17
41, 209, 157169, 213321, 303093, 382449, 2145353, 2478785 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Table of n, a(n) for n=1..8.
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
MATHEMATICA
lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 209, 2}]; lst
PROG
(PARI) 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
CROSSREFS
Subsequence of A002253.
Cf. A000215, A039687, A057775, A057778, A201364, A226366.
KEYWORD
nonn,hard,more
AUTHOR
Arkadiusz Wesolowski, Jan 17 2012
STATUS
approved
A007505 Primes of form 3*2^n - 1.
(Formerly M1395) 		+10
15
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407, 59421121885698253195157962751, 30423614405477505635920876929023 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
REFERENCES
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).
LINKS
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
FORMULA
a(n) = 3*2^A002235(n)-1. - Zak Seidov, Jul 21 2016
MATHEMATICA
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 *)
PROG
(Magma) [a: n in [0..200] | IsPrime(a) where a is 3*2^n-1]; // Vincenzo Librandi, Mar 20 2013
(Haskell)
a007505 n = a007505_list !! (n-1)
a007505_list = filter ((== 1) . a010051') a083329_list
-- Reinhard Zumkeller, Sep 10 2013
(PARI) for(n=0, 100, if(isprime(t=3<<n-1), print1(t", "))) \\ Charles R Greathouse IV, Feb 07 2017
CROSSREFS
Subsequence of A083329.
See A002235 for more terms.
Cf. A039687 (primes of the form 3*2^n+1).
Cf. A010051.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Robert G. Wilson v
STATUS
approved
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. +10
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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, ..., .
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..53
PROG
(PARI) isA334092(n) = (isprime(n)&&2==A329697(n));
(PARI)
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)));
(PARI) 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)<lim, if(isprime(t+1), listput(v, t+1)))); Set(v) \\ Charles R Greathouse IV, Apr 14 2020
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Apr 14 2020
EXTENSIONS
More terms from Giovanni Resta, Apr 14 2020
STATUS
approved
A074781 Primes of the form p*2^k + 1 for any k and any prime p. +10
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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)
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.
LINKS
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.
EXAMPLE
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.
MAPLE
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
MATHEMATICA
Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)
CROSSREFS
Cf. A000079, A006093, A006530, A052126, A339464.
Cf. other ratios : A339463, A339465, A339466.
Subsequences: A039687, A051900, A058500 (this sequence without the Fermat primes), A090866, A147545,
KEYWORD
nonn
AUTHOR
Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
STATUS
approved
A050526 Primes of form 5*2^n+1. +10
9
11, 41, 641, 40961, 163841, 167772161, 2748779069441, 180143985094819841, 188894659314785808547841, 193428131138340667952988161, 850705917302346158658436518579420528641 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms are odd since if n is even, then 5*2^n+1 is divisible by 3. - Michele Fabbrini, Jun 06 2021
LINKS
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
FORMULA
a(n) = A083575(A002254(n)). - Michel Marcus, Mar 29 2018
MAPLE
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
PROG
(Magma) [a: n in [1..200] | IsPrime(a) where a is 5*2^n + 1]; // Vincenzo Librandi, Mar 06 2018
(GAP) Filtered(List([1..270], n->5*2^n + 1), IsPrime); # Muniru A Asiru, Mar 06 2018
(PARI) lista(nn) = {for(k=1, nn, if(ispseudoprime(p=5*2^k+1), print1(p, ", "))); } \\ Altug Alkan, Mar 29 2018
CROSSREFS
For the corresponding exponents n see A002254.
Cf. A019434, A039687, A050527, A050528, A050529, A195745, A300406, A300407, A300408.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 29 1999
STATUS
approved
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