A045718 -id:A045718 - cp's OEIS frontend

A194591 Least k >= 0 such that n2^k - 1 or n2^k + 1 is prime, or -1 if no such value exists.

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 5, 0, 3, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 1

Offset: 1

Arkadiusz Wesolowski, Aug 29 2011

sign

Fred Cohen and J. L. Selfridge showed that a(n) = -1 infinitely often.

a(n) = 0 iff n is in A045718.

			For n=7, 7*2^0-1 and 7*2^0+1 are composite, but 7*2^1-1=13 is prime, so a(7)=1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp. 29 (1975), 79-81.
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639.
Cf. A040081, A040076, A076335, A180247.
Cf. A217892 and A194600 (indices and values of the records).

Table[k = 0; While[! PrimeQ[n*2^k - 1] && ! PrimeQ[n*2^k + 1], k++]; k, {n, 100}] (* T. D. Noe, Aug 29 2011 *)

If a(n)>0, then a(2n)=a(n)-1.

A079364 Composite numbers having two composite neighbors.

Original entry on oeis.org

9, 15, 21, 25, 26, 27, 33, 34, 35, 39, 45, 49, 50, 51, 55, 56, 57, 63, 64, 65, 69, 75, 76, 77, 81, 85, 86, 87, 91, 92, 93, 94, 95, 99, 105, 111, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 129, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 153, 154, 155

Offset: 1

Reinhard Zumkeller, Feb 15 2003

nonn

In other words, composite numbers that are not nearest-neighbors of primes. - Omar E. Pol, Jan 02 2009

Complement of A210940. - Omar E. Pol, Apr 18 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A002808, A014574, A045718.

Cf. A010051, A221309 (subsequence).

a079364 n = a079364_list !! (n-1)
a079364_list = filter
   (\x -> a010051' (x - 1) == 0 && a010051' (x + 1) == 0) a002808_list
-- Reinhard Zumkeller, Jan 10 2013

Select[Range[6! ],!PrimeQ[ # ]&&!PrimeQ[ #-1]&&!PrimeQ[ #+1]&] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2010 *)
With[{r=Complement[Range[160],Prime[Range[PrimePi[160]]]]}, Transpose[ Select[ Partition[r,3,1], Differences[#]=={1,1}&]][[2]]] (* Harvey P. Dale, Feb 05 2012 *)
Mean/@SequencePosition[Table[If[CompositeQ[n],1,0],{n,200}],{1,1,1}] (* Harvey P. Dale, May 10 2025 *)

A084925 Inverse hyperbolic cotangent irreducible numbers: positive integers such that the arccoth of these numbers form a basis for the space of arccoth of rationals >=1. The hyperbolic analog of the Stormer numbers (A005528).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 66, 68, 70, 72, 74, 78, 80, 82, 84, 88, 90, 92, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 122, 126, 128, 130, 132, 136, 138, 140, 142, 144, 148, 150

Offset: 1

Paul D. Hanna, Jun 12 2003

nonn

n is in the sequence if y = (xn+1)/(x+n) is noninteger for all integer x where 1 < x < n. Equivalently, n is in the sequence when n cannot be formed by (xy-1)/(x-y) for all integers x and y where x < n and 1 < y < x, so n cannot satisfy ((n+1)/(n-1))*((x+1)/(x-1)) = ((y+1)/(y-1)). Thus all the nearest neighbors of the primes (A045718) appear in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul D. Hanna)

Cf. A005528 (Stormer numbers), A045718, A084926.

for(n=1,150,x=1; b=0; while(x=(x *n+1),b=b+1)); if(b<=0,print1(n,",")))

A100319 Even numbers m such that at least one of m-1 and m+1 is composite.

Original entry on oeis.org

8, 10, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 140, 142, 144, 146, 148

Offset: 1

Rick L. Shepherd, Nov 13 2004

nonn

Subsequence of A100318. For each k >= 0, a(k+1) = a(k) + 2 unless a(k) + 1 and a(k) + 3 are twin primes, in which case a(k+1) = a(k) + 4 (as a(k) - 1 and a(k) + 5 are divisible by 3).

The even nonisolated primes(n+1). - Juri-Stepan Gerasimov, Nov 09 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A100318 (supersequence containing odd and even n), A045718 (n such that at least one of n-1 and n+1 is prime).
Cf. A167692(the even nonisolated nonprimes). - Juri-Stepan Gerasimov, Nov 09 2009
Cf. A005818, A038179, A007310, A038511, A025584.
Complement of A014574 (average of twin prime pairs) w.r.t. A005843 (even numbers), except for missing term 2.

Select[2*Range[100], CompositeQ[#-1] || CompositeQ[#+1] &]  (* G. C. Greubel, Mar 09 2019 *)

forstep(n=4,300,2,if(isprime(n-1)+isprime(n+1)<=1,print1(n,",")))

[n for n in (3..250) if mod(n,2)==0 and (is_prime(n-1) + is_prime(n+1)) < 2] # G. C. Greubel, Mar 09 2019

a(n) = A167692(n+1). - Juri-Stepan Gerasimov, Nov 09 2009

A147819 Nearest-neighbors of odd primes.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 36, 38, 40, 42, 44, 46, 48, 52, 54, 58, 60, 62, 66, 68, 70, 72, 74, 78, 80, 82, 84, 88, 90, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 126, 128, 130, 132, 136, 138, 140, 148, 150, 152, 156, 158, 162

Offset: 1

Omar E. Pol, Nov 14 2008

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000040, A045718, A065091, A134924, A134930.

#+{-1,1}&/@Prime[Range[2,60]]//Flatten//Union (* Harvey P. Dale, Jun 13 2017 *)

A088259 Perfect powers which have at least one prime neighbor.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 100, 128, 196, 256, 400, 576, 676, 1296, 1600, 2916, 3136, 4356, 5476, 7056, 8100, 8192, 8836, 12100, 13456, 14400, 15376, 15876, 16900, 17956, 21316, 22500, 24336, 25600, 28900, 30976, 32400, 33856, 41616, 42436, 44100

Offset: 1

Amarnath Murthy, Sep 27 2003

nonn

If K is a term and K-1 is the neighboring prime then it must be a Mersenne prime.

Conjecture: sequence is infinite.

Harvey P. Dale, Table of n, a(n) for n = 1..322 (all terms up to 10 million)

Intersection of A001597 and A045718.

Cf. A088256, A088257, A088258.

Join[{1},Select[Range[45000],GCD@@FactorInteger[#][[All,2]]>1 && AnyTrue[ #+{1,-1},PrimeQ]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2020 *)

Corrected and extended by Ray Chandler, Sep 28 2003

Offset changed by Andrew Howroyd, Sep 22 2024

A134924 Nearest-neighbors of isolated primes.

Original entry on oeis.org

1, 3, 22, 24, 36, 38, 46, 48, 52, 54, 66, 68, 78, 80, 82, 84, 88, 90, 96, 98, 112, 114, 126, 128, 130, 132, 156, 158, 162, 164, 166, 168, 172, 174, 210, 212, 222, 224, 232, 234, 250, 252, 256, 258, 262, 264, 276, 278, 292, 294, 306, 308

Offset: 1

Omar E. Pol, Nov 27 2007

Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Cf. A007510, A045718, A134926, A134930.

ip=Select[Prime[Range[65]],NoneTrue[{#-2,#+2},PrimeQ]&];Union[Flatten[Table[n+{1, -1}, {n, ip}]]] (* James C. McMahon, Apr 12 2025 *)

A210940 The prime numbers and their nonprime nearest-neighbors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 28, 29, 30, 31, 32, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 66, 67, 68, 70, 71, 72, 73, 74, 78, 79, 80, 82, 83, 84, 88, 89, 90, 96, 97, 98, 100

Offset: 1

Omar E. Pol, Apr 17 2012

nonn

The prime numbers and their nearest-neighbors without repetitions.

Union of A210939 and A000040. Complement of A079364.

Cf. A018252, A045718, A134928, A134930.

{#-1,#,#+1}&/@Prime[Range[30]]//Flatten//Union (* Harvey P. Dale, Jul 06 2019 *)

Corrected (74 added) by Harvey P. Dale, Jul 06 2019

A210939 Nonprime nearest-neighbors of the primes.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 36, 38, 40, 42, 44, 46, 48, 52, 54, 58, 60, 62, 66, 68, 70, 72, 74, 78, 80, 82, 84, 88, 90, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 126, 128, 130, 132, 136, 138, 140, 148, 150, 152, 156

Offset: 1

Omar E. Pol, Apr 17 2012

nonn

Essentially the same as A147819. R. J. Mathar, Jun 25 2012

Nonprimes in A045718.

Cf. A000040, A018252, A134928, A134930, A210940.

Select[Range[156], ! PrimeQ[#] && (PrimeQ[# - 1] || PrimeQ[# + 1]) &] (* T. D. Noe, Apr 18 2012 *)
Join[{1},Flatten[#+{-1,1}&/@Prime[Range[3,40]]]//Union] (* Harvey P. Dale, Oct 22 2022 *)

A128510 Composites c such that c*A001414(c) is adjacent to a prime.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 16, 20, 21, 25, 26, 28, 33, 34, 35, 36, 38, 40, 42, 44, 46, 50, 51, 52, 54, 55, 56, 60, 64, 65, 68, 72, 74, 76, 80, 81, 82, 85, 90, 93, 95, 96, 98, 100, 102, 110, 111, 115, 119, 121, 122, 123, 124, 126, 132, 133, 135, 138, 140, 143, 144, 145, 146, 148, 150

Offset: 1

J. M. Bergot, May 07 2007

The composites c of A002808 are multiplied by the sum of their prime factors (with multiplicity), and are placed into the sequence if that product is in A045718.

			c = 52= A002808(74) has prime factor sum A001414(52) = 17, and 52*17 = 883+1 is one away from the prime 883, which adds 52 to the sequence.

Cf. A066073.

A001414 := proc(n) local fcts,d ; fcts := ifactors(n)[2] ; add(op(1,d)*op(2,d),d=fcts) ; end proc:
isA045718 := proc(n) isprime(n+1) or isprime(n-1) ; end proc:
isA128510 := proc(n) local c; if not isprime(n) then c := n*A001414(n) ; isA045718(c) ; else false; end if ; end proc:
for n from 4 to 500 do if isA128510(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Nov 02 2009

8 inserted and sequence extended by R. J. Mathar, Nov 02 2009

cp's OEIS Frontend

A194591 Least k >= 0 such that n*2^k - 1 or n*2^k + 1 is prime, or -1 if no such value exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A079364 Composite numbers having two composite neighbors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A084925 Inverse hyperbolic cotangent irreducible numbers: positive integers such that the arccoth of these numbers form a basis for the space of arccoth of rationals >=1. The hyperbolic analog of the Stormer numbers (A005528).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A100319 Even numbers m such that at least one of m-1 and m+1 is composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A147819 Nearest-neighbors of odd primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A088259 Perfect powers which have at least one prime neighbor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A134924 Nearest-neighbors of isolated primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A210940 The prime numbers and their nonprime nearest-neighbors.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A210939 Nonprime nearest-neighbors of the primes.

Views

A194591 Least k >= 0 such that n2^k - 1 or n2^k + 1 is prime, or -1 if no such value exists.