A181492 -id:A181492 - 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

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

A181491 Primes of the form p = 3*2^k - 1 such that p+2 is also prime.

Original entry on oeis.org

5, 11, 191, 786431

Offset: 1

M. F. Hasler, Oct 30 2010

Sequence A181490 lists the exponents k, sequences A181492 and A181493 the corresponding upper twin prime and their average.

a(5) > 3 * 2 ^ 3000 + 1. - Max Z. Scialabba, Dec 24 2023

Cf. A001097, A001359, A006512, A014574.

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

A181491 = A007283 intersect A014574 = A181492 - 2 = A181493 - 1 = 3*2^A153890 - 1.

A181493 Numbers of the form 3*2^k which are the average of twin primes, i.e., a(n)-1 and a(n)+1 are both prime.

Original entry on oeis.org

6, 12, 192, 786432

Offset: 1

M. F. Hasler, Oct 30 2010

Sequence A181490 lists the exponents k, sequences A181491 and A181492 the corresponding twin primes.

Cf. A001097, A001359, A006512, A014574.

Select[3 2^Range[500],PrimeQ[#-1]&&PrimeQ[#+1]&]  (* Harvey P. Dale, Jan 18 2011 *)

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

A181493 = A014574 intersect A007283 = A181491 + 1 = A181492 - 1 = 3*2^A181490.
A181493(n) = A007283(A181490(n)).
A181493 = A014574 intersect A007283 = A181491 + 1 = A181492.
A181493(n) = A007283(A181490(n)).

A276136 Numbers m > 1 such that the largest odd divisors of m-1, m, and m+1 are prime.

Original entry on oeis.org

6, 11, 12, 13, 23, 47, 192, 193, 383, 786432

Offset: 1

Juri-Stepan Gerasimov, Aug 22 2016

nonn

Conjecture: this sequence is finite.
Any further terms are greater than 10^11. - Charles R Greathouse IV, Aug 22 2016
From Robert Israel, Apr 27 2020: (Start)
Each term is either of the form 3*2^k with 3*2^k-1 and 3*2^k+1 prime, or 3*2^k-1 with 3*2^k-1 prime and 3*2^(k-1)-1 prime, or 3*2^k+1 with 3*2^k+1 prime and 3*2^(k-1)+1 prime.
Any further terms > 10^2000.
(End)

			6 is in this sequence because the largest odd divisor of 5 is 5, the largest odd divisor of 6 is 3 and the largest odd divisor of 7 is 7, and all three are prime.

Supersequence of A181493. Subsequence of A038550.

Cf. A001227, A181490, A181491, A181492, A275418.

[n: n in [2..3000000] | NumberOfDivisors(2*(n-1))- NumberOfDivisors(n-1)eq 2 and NumberOfDivisors(2(n))-NumberOfDivisors(n) eq 2 and NumberOfDivisors(2*(n+1))- NumberOfDivisors(n+1) eq 2];

Res:= 6:
for k from 2  while length(3*2^k-1)<1000 do
  if (isprime(3*2^k-1) and isprime(3*2^(k-1)-1)) then Res:= Res, 3*2^k-1
    fi;
  if (isprime(3*2^k-1) and isprime(3*2^k+1)) then Res:= Res, 3*2^k;
    fi;
  if (isprime(3*2^k+1) and isprime(3*2^(k-1)+1)) then Res:= Res, 3*2^k+1;
    fi;
od:
Res; # Robert Israel, Apr 27 2020

Select[Range[2, 10^6], Function[n, Times @@ Boole@ PrimeQ@ Map[First@ Reverse@ DeleteCases[Divisors@ #, d_ /; EvenQ@ d] &, n + Range[-1, 1]] == 1]] (* Michael De Vlieger, Aug 22 2016 *)
SequencePosition[Table[If[PrimeQ[Max[Select[Divisors[n],OddQ]]],1,0],{n,800000}],{1,1,1}][[;;,1]]+1 (* Harvey P. Dale, Jun 27 2023 *)

isA038550(n)=isprime(n>>valuation(n,2))
is(n)=isA038550(n-1) && isA038550(n) && isA038550(n+1) \\ Charles R Greathouse IV, Aug 22 2016

forprime(p=2,1e11, my(a=isA038550(p-1),b=isA038550(p+1)); if(a && isA038550(p-2), print1(p-1", ")); if(a && b, print1(p", ")); if(b && isA038550(p+2), print1(p+1", "))) \\ may print numbers several times, but won't skip numbers; Charles R Greathouse IV, Aug 22 2016

A038550(a(n-1)) + 1 = A038550(a(n)) = A038550(a(n+1)) - 1.

a(n) >> n log n. - Charles R Greathouse IV, Aug 22 2016

A092680 Numbers of the form 3*2^k with a single anti-divisor.

Original entry on oeis.org

3, 6, 96, 393216

Offset: 1

Lior Manor, Mar 03 2004

See A066272 for definition of anti-divisor.

If it exists, a(5) > 3*2^(1000). See A092679. - J.W.L. (Jan) Eerland, Nov 13 2022

Cf. A066272, A066466, A092679, A181490, A181491, A181492.

from itertools import count, islice
from sympy.ntheory.factor_ import antidivisor_count
def A092680_gen(): return filter(lambda n: antidivisor_count(n)==1,(3*2**k for k in count(0)))
A092680_list = list(islice(A092680_gen(),4)) # Chai Wah Wu, Jan 04 2022

a(n) = 3*2^A092679(n).

a(n) = 3*2^(A181490(n)-1) = (A181491(n)+1)/2 = (A181492(n)-1)/2. - Max Alekseyev, Feb 14 2025

A181494 Twin primes (A001097) of the form 3*2^k +- 1.

Original entry on oeis.org

5, 7, 11, 13, 191, 193, 786431, 786433

Offset: 1

M. F. Hasler, Oct 30 2010

Cf. A001097, A001359, A006512, A014574.

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

A181494 = A181491 union A181492.

A181494(n) = A181493(ceiling(n/2)) + (-1)^n = A007283(A181490(ceiling(n/2))) + (-1)^n.

cp's OEIS Frontend

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

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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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A181491 Primes of the form p = 3*2^k - 1 such that p+2 is also prime.

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A181493 Numbers of the form 3*2^k which are the average of twin primes, i.e., a(n)-1 and a(n)+1 are both prime.

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Mathematica

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A276136 Numbers m > 1 such that the largest odd divisors of m-1, m, and m+1 are prime.

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Magma

Maple

Mathematica

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A092680 Numbers of the form 3*2^k with a single anti-divisor.

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A181494 Twin primes (A001097) of the form 3*2^k +- 1.

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PARI

Formula

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