A050415 -id:A050415 - cp's OEIS frontend

A090670 Odd numbers k such that 2k-3 is a prime of the form 4j+3.

3, 5, 7, 11, 13, 17, 23, 25, 31, 35, 37, 41, 43, 53, 55, 65, 67, 71, 77, 83, 85, 91, 97, 101, 107, 113, 115, 121, 127, 133, 137, 143, 155, 157, 167, 175, 181, 185, 191, 193, 211, 217, 221, 223, 233, 235, 241, 245, 247, 251, 253, 263, 275, 283, 287, 295, 301, 305

Offset: 1

Giovanni Teofilatto, Dec 17 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A087915, A002145.

Cf. A050415 (primes of the form 2^k-3).

okQ[n_]:=Module[{x=2n-3},PrimeQ[x]&&IntegerQ[(x-3)/4]]; Select[Range[1,315,2],okQ]  (* Harvey P. Dale, Jan 12 2011 *)

a(n) = A087915(n)+3.

Offset changed from 0 to 1 by Vincenzo Librandi, Dec 13 2011

A093810 Smallest prime factor of 2^n-3.

Original entry on oeis.org

1, 5, 13, 29, 61, 5, 11, 509, 1021, 5, 4093, 19, 16381, 5, 13, 53, 11, 5, 1048573, 773, 4194301, 5, 16777213, 479, 37, 5, 11, 536870909, 23, 5, 9241, 29, 5113, 5, 242819, 47189, 11, 5, 13, 23, 47, 5, 5927, 2087, 227, 5, 11, 19, 59, 5, 13, 2203, 36217, 5, 181

Offset: 2

Yasutoshi Kohmoto, May 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 2..626 (terms 2..200 from Vincenzo Librandi)

Cf. A020639, A036563, A050414, A050415, A093817.

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[2^n - 3][[1]], {n, 2, 60}] (* Robert G. Wilson v, May 24 2004 *)
FactorInteger[#][[1,1]]&/@(2^Range[2,60]-3) (* Harvey P. Dale, Aug 21 2016 *)

a(n) = A020639(A036563(n)). - Amiram Eldar, Sep 12 2022

More terms from Robert G. Wilson v, May 24 2004

A093817 Largest prime factor of 2^n-3.

Original entry on oeis.org

1, 5, 13, 29, 61, 5, 23, 509, 1021, 409, 4093, 431, 16381, 6553, 71, 2473, 23831, 97, 1048573, 2713, 4194301, 1677721, 16777213, 70051, 5197, 31033, 1877171, 536870909, 46684427, 22605091, 464773, 296204641, 3360037, 6871947673, 283007

Offset: 2

Yasutoshi Kohmoto, May 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 2..626 (terms 2..200 from Vincenzo Librandi)

Cf. A006530, A036563, A050414, A050415, A093810.

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[2^n - 3][[ -1]], {n, 2, 46}] (* Robert G. Wilson v, May 24 2004 *)
Table[FactorInteger[2^n-3][[-1,1]],{n,2,40}] (* Harvey P. Dale, Feb 01 2015 *)

a(n) = A006530(A036563(n)). - Amiram Eldar, Sep 12 2022

More terms from Robert G. Wilson v, May 24 2004

A067932 Primes p such that p+3 == 0 (mod phi(p+3)).

Original entry on oeis.org

3, 5, 13, 29, 61, 509, 1021, 4093, 16381, 1048573, 4194301, 16777213, 536870909, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301

Offset: 1

Benoit Cloitre, Feb 22 2002

phi(n) divides n iff n=1 or n=2^w*3^u for w>=1 and u>=0 (see A007694). Such an n can only have the form p+3 if n=6 or n is a power of 2. So the terms of the sequence are 3 and the primes of the form 2^n-3, listed in A050415.

Vincenzo Librandi, Table of n, a(n) for n = 1..30

Prepend[Select[2^Range[2, 200]-3, PrimeQ], 3]

Edited and extended by Robert G. Wilson v, Feb 27 2002 and by Dean Hickerson, Mar 21 2002

A167917 Mersenne primes that belong to Cunningham chains = {3, 7} U {Mp | 2Mp - 1 is prime. (Mp a Mersenne prime)}.

Original entry on oeis.org

3, 7, 31, 8191, 524287

Offset: 1

Washington Bomfim, Nov 15 2009

If p is prime, p >= 5, and Mp belongs to a chain, Mp is always the first term of a chain of the second kind. This is true since (Mp+1)/2 = (2^p - 1 +1)/2 = 2^(p-1), which is composite for p >= 3. (Mp-1)/2 = (2^p - 1 -1)/2 = 2^(p-1)-1 = a. For p >= 5, a is composite since a>3, and a mod 3 = 0. Finally 2Mp + 1 = 2(2^p - 1)+1 = 2^(p+1)-1 = a. If p>=3, a is composite because a > 3, and a mod 3 = 0. We can conclude that beginning with 31, a Mersenne prime can only starts a Cunningham chain of the second kind. If Mp >= 31 starts a chain, the second term of this chain is 2Mp -1=2(2^p - 1)-1 = 2^(p+1) - 3.
That is a number of the form 2^N - 3, even N, so also of the form a^2 - 3, a = 2^(N/2). In this case any factor f of the second term of a chain satisfies f mod 24=1, or f mod 24=11, or f mod 24=13, or f mod 24=23. (1) The next term of this sequence is an unknown Mersenne prime. Probably many primes of this kind will be determined until this term be found. In the work with the known Mersenne primes, M42643801 gives T=2^(42643801+1) -3. The smallest factor of T is f = 38334482051, which is greater than 2^35.
Considering the probabilities given in the second reference, one can conclude that before T was identified as composed (by the exam of all the primes less than f satisfying (1)), the probability of prime T reached a value of 1 in 609,197. This probability is small, but not negligible. Note that the largest known Cunningham chain of length 2 has starting prime 607095* 2^176311 - 1. This is a "very small chain" compared with a chain beginning with a new Mersenne prime.

			a(1) = 3 since 2*3 - 1 = 5. a(2) = 7 because 2*7 - 1 = 13.

Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
Mathpages, Some Properties of the Lucas Sequence
GIMPS, Mathematics and Research Strategy
Wikipedia, Cunningham chain

Cf. A000043, A000668, A050415, A050414.

A172041 Primes of the form 2^p-3 with p also prime.

Original entry on oeis.org

5, 29, 536870909, 13803492693581127574869511724554050904902217944340773110325048447598589

Offset: 1

Vincenzo Librandi, Jan 24 2010

nonn

Next two terms are 2^42689-3 and 2^69337-3.
Subsequence of A050415.
Corresponding p are the primes in A050414.
The next term has 12851 digits. - Harvey P. Dale, Aug 27 2023

Select[Table[2^p-3,{p,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Aug 27 2023 *)

Definition clarified by R. J. Mathar, Jan 28 2010

A347476 Numbers which give a prime number when 0's and 1's are interchanged in their binary representation.

Original entry on oeis.org

4, 5, 8, 10, 12, 13, 18, 20, 24, 26, 28, 29, 32, 34, 40, 44, 46, 50, 52, 56, 58, 60, 61, 66, 68, 74, 80, 84, 86, 90, 96, 98, 104, 108, 110, 114, 116, 120, 122, 124, 125, 128, 142, 146, 148, 152, 154, 158, 166, 172, 176, 182, 184, 188, 194, 196, 202, 208, 212

Offset: 1

Srijan Suryansh, Sep 03 2021

Subsequence of primes is A050415 and then, the obtained prime is always 2. We have: prime p = 2^k-3, k>=3 -> p = 11...1101_2 with a string of (k-2) digits 1 before '01' ==> 00...0010_2 = 10_2 -> 2_10. - Bernard Schott, Oct 21 2021

Cf. A000040, A035327, A050415.

q:= n-> isprime(Bits[Nand](n$2)):
select(q, [$1..300])[];  # Alois P. Heinz, Sep 03 2021

Select[Range[200], PrimeQ @ FromDigits[IntegerDigits[#, 2] /. {0 -> 1, 1 -> 0}, 2] &] (* Amiram Eldar, Sep 03 2021 *)

isok(m) = isprime(fromdigits(apply(x->1-x, binary(m)), 2)); \\ Michel Marcus, Sep 03 2021

is(n) = isprime(1<<(logint(n, 2) + 1) - n - 1) \\ David A. Corneth, Sep 03 2021

from sympy import isprime
def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def ok(n): return isprime(int(comp(bin(n)[2:]), 2))
print(list(filter(ok, range(214)))) # Michael S. Branicky, Sep 03 2021

More terms from Michael S. Branicky, Sep 03 2021

A267943 Numbers n such that 2^n - 3 and 3*2^n - 1 are both prime.

Original entry on oeis.org

3, 4, 6, 94

Offset: 1

Arkadiusz Wesolowski, Jan 22 2016

The intersection of A002235 and A050414 is not empty (3 does not belong to A267985).

			a(3) = 6 because 2^6 - 3 = 61 and 3*2^6 - 1 = 191 are both prime.

Cf. A002235, A007505, A050414, A050415, A238694, A267985.

[n: n in [2..94] | IsPrime(2^n-3) and IsPrime(3*2^n-1)];

isok(n) = isprime(2^n-3) && isprime(3*2^n-1);

A002235 INTERSECT A050414.

cp's OEIS Frontend

A090670 Odd numbers k such that 2*k-3 is a prime of the form 4*j+3.

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A093810 Smallest prime factor of 2^n-3.

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A093817 Largest prime factor of 2^n-3.

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A067932 Primes p such that p+3 == 0 (mod phi(p+3)).

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A167917 Mersenne primes that belong to Cunningham chains = {3, 7} U {Mp | 2Mp - 1 is prime. (Mp a Mersenne prime)}.

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A172041 Primes of the form 2^p-3 with p also prime.

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A347476 Numbers which give a prime number when 0's and 1's are interchanged in their binary representation.

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A267943 Numbers n such that 2^n - 3 and 3*2^n - 1 are both prime.

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A090670 Odd numbers k such that 2k-3 is a prime of the form 4j+3.