Search: a039687 -id:a039687
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3, 12, 48, 96, 384, 6144, 393216, 805306368, 103079215104, 549755813888, 110680464442257309696, 1176956575385002643219210516851437453019191645837006471168
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OFFSET
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1,1
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COMMENTS
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Here we corrected a(10) which was 1649267441664 in Golomb's article. - R. J. Mathar, Jul 14 2015
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LINKS
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MAPLE
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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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
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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MATHEMATICA
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PROG
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CROSSREFS
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Essentially a duplicate of A004119.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A002253
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Numbers k such that 3*2^k + 1 is prime.
(Formerly M1318 N0506)
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+10
22
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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
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OFFSET
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1,2
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COMMENTS
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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)
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REFERENCES
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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).
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LINKS
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FORMULA
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PROG
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(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
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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EXTENSIONS
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Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 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
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STATUS
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approved
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A004119
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a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.
(Formerly M3308)
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+10
19
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
O.g.f.: -(-1-x+3*x^2)/((2*x-1)*(x-1)). - R. J. Mathar, Nov 23 2007
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MAPLE
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MATHEMATICA
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Prepend[Table[3*2^n + 1, {n, 0, 32}], 1] (* or *)
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PROG
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(Magma) [1] cat [n le 1 select 4 else 2*Self(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015
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CROSSREFS
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A181565 is an essentially identical sequence.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A081091
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Primes of the form 2^i + 2^j + 1, i > j > 0.
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+10
19
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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Select[Flatten[Table[2^i + 2^j + 1, {i, 21}, {j, i-1}]], PrimeQ] (* Alonso del Arte, Jan 11 2011 *)
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PROG
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(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) 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)
(Haskell)
a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
(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'))))
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CROSSREFS
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Cf. A095077 (primes with four bits set).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A204620
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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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+10
17
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 209, 2}]; lst
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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A007505
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Primes of form 3*2^n - 1.
(Formerly M1395)
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+10
15
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2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407, 59421121885698253195157962751, 30423614405477505635920876929023
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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MATHEMATICA
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Reap[For[n = 0, n <= 103, n++, If[PrimeQ[p = 3*2^n - 1], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2012 *)
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PROG
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(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
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CROSSREFS
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Cf. A039687 (primes of the form 3*2^n+1).
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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A334092
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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.
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+10
14
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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
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OFFSET
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1,1
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COMMENTS
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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, ..., .
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LINKS
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PROG
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(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));
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A074781
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Primes of the form p*2^k + 1 for any k and any prime p.
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+10
11
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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
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OFFSET
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1,1
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COMMENTS
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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)
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.
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LINKS
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EXAMPLE
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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.
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MAPLE
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alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:
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MATHEMATICA
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Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
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STATUS
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approved
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A050526
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Primes of form 5*2^n+1.
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+10
9
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11, 41, 641, 40961, 163841, 167772161, 2748779069441, 180143985094819841, 188894659314785808547841, 193428131138340667952988161, 850705917302346158658436518579420528641
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OFFSET
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1,1
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COMMENTS
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All terms are odd since if n is even, then 5*2^n+1 is divisible by 3. - Michele Fabbrini, Jun 06 2021
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LINKS
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FORMULA
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MAPLE
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a:=(n, k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
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PROG
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(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
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CROSSREFS
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For the corresponding exponents n see A002254.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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