A099047 -id:A099047 - cp's OEIS frontend

A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

13, 17, 25, 28, 32, 38, 43, 46, 47, 58, 59, 60, 61, 62, 67, 71, 72, 73, 77, 80, 85, 88, 92, 93, 94, 101, 102, 103, 104, 107, 108, 109, 110, 118, 122, 123, 124, 127, 130, 133, 137, 143, 144, 145, 148, 149, 150, 151, 152, 160, 161, 162, 163, 164, 167, 170, 171, 172

Offset: 1

Alexandre Wajnberg, Apr 17 2005

Complement of A147820. - Omar E. Pol, Nov 17 2008

m is in the sequence iff A177961(m)Vladimir Shevelev, May 16 2010

			a(1)=13 is the first number satisfying simultaneously the two rules.

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

Intersection of A047845 and A104275.
Cf. A040040, A147820, A025583.
Cf. A010051, A002808, A099047.

a104278 n = a104278_list !! (n-1)
a104278_list = [m | m <- [1..],
                    a010051' (2 * m - 1) == 0 && a010051' (2 * m + 1) == 0]
-- Reinhard Zumkeller, Aug 04 2015

Select[ Range[300], !PrimeQ[2# + 1] && !PrimeQ[2# - 1] &] (* Robert G. Wilson v, Apr 18 2005 *)
Select[Range[300],NoneTrue[2#+{1,-1},PrimeQ]&] (* The program uses the NoneTrue function from Mathematica version 10 *)  (* Harvey P. Dale, Jul 07 2015 *)

select( {is_A104278(n)=!isprime(2*n-1)&&!isprime(2*n+1)}, [1..222]) \\ M. F. Hasler, Apr 29 2024

a(n) = (A025583-1)/2. - Bill McEachen, Feb 05 2025

More terms from Robert G. Wilson v, Apr 18 2005

A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

Original entry on oeis.org

120, 144, 186, 204, 216, 246, 288, 300, 324, 342, 414, 426, 474, 516, 528, 534, 552, 582, 624, 636, 666, 696, 714, 780, 792, 804, 816, 834, 846, 870, 894, 900, 924, 960, 1002, 1026, 1044, 1056, 1074, 1080, 1134, 1140, 1146, 1158, 1176, 1206, 1242, 1254, 1266, 1272, 1314, 1332, 1338, 1344, 1350

Offset: 1

Antonio Gimenez, Jul 05 2015

nonn

From Brian Almond, Jun 23 2020: (Start)
For every prime gap g, there is a run of consecutive a(n) of length max{[(g+2)/6]-1,0}.
Gaps between successive a(n) correspond to clusters of primes all within +- 8 of each other.  The number of primes within a gap G = a(n+1) - a(n) ranges from (G/6 - 1) to (G/6 - 1) plus the number of twin primes within the gap.
Record gaps in a(n) are 24 at a(1)=120, 42 at a(2)=144, 72 at a(10)=342 and 84 at a(1003)=14706 (the next gaps of 84 occur at a(43136164)=369008652 and a(643519601)=5244999552).  No larger record gaps exist below 10^10 (n <= 1239026836).
(End)
Define a "small-gap k-tuple" to be an admissible k-tuple with all of its gaps in {2,4,6,8}.  Every gap G = a(n+1) - a(n) >= 18 contains a small-gap k-tuple with k >= G/6 - 1 and diameter G-14, G-12 or G-10.  For example, at n=40 the gap between 1080 and 1134 contains the 9-tuple p+{0,4,6,10,16,22,30,36,42} for p=1087. - Brian Almond, Jul 25 2020

			For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Intersection of A008588 and A099047. - Michel Marcus, Jul 06 2015

Cf. A060461.

[n: n in [6..2000 by 6] | not IsPrime(n-1) and not IsPrime(n+1)]; // Vincenzo Librandi, Jul 08 2015

Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* Harvey P. Dale, May 21 2017 *)

select(x->!isprime(x-1)&&!isprime(x+1), vector(10^3,j,6*j) ) \\ Joerg Arndt, Jul 06 2015

a(n) = 6 * A060461(n). - Brian Almond, Jun 22 2020

A285805 Prime numbers p such that 6p-1 and 6p+1 are composite numbers.

Original entry on oeis.org

31, 41, 71, 79, 89, 97, 139, 149, 167, 179, 191, 193, 211, 223, 251, 281, 307, 337, 349, 353, 401, 409, 419, 421, 431, 433, 479, 487, 491, 499, 509, 521, 541, 547, 563, 571, 587, 619, 631, 643, 659, 673, 677, 691, 701, 719, 739, 757, 769, 809

Offset: 1

Dimitris Valianatos, Apr 26 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060212, A099047.

filter:= n -> isprime(n) and not isprime(6*n-1) and not isprime(6*n+1):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jan 05 2020

Select[Prime@Range@150, ! PrimeQ[6 # - 1] && ! PrimeQ[6 # + 1] &] (* Robert G. Wilson v, Apr 27 2017 *)
Select[Prime[Range[150]],NoneTrue[6#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Jun 10 2022 *)

{forprime(n=3, 1000, if(!isprime(6*n-1)&&!isprime(6*n+1), print1(n", ")))}

A174392 Neither n-1 nor n+1 is prime.

Original entry on oeis.org

0, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 116, 117, 118

Offset: 1

Juri-Stepan Gerasimov, Mar 18 2010

Union of {0} and A099047. [R. J. Mathar, Aug 09 2010]

			a(1)=0 because 0-1=-1 and 0+1=1 are nonprime; a(2)=5 because 5-1=4 and 5+1=6 are nonprime.

Cf. A099047.

Select[Range[0,150],!PrimeQ[#-1]&&!PrimeQ[#+1]&] (* Harvey P. Dale, Aug 15 2012 *)
Join[{0},Mean/@SequencePosition[Table[If[PrimeQ[n],0,1],{n,200}],{1,x_,1}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 17 2019 *)

Corrected by Charles R Greathouse IV, Mar 18 2010

Incorrect comment removed by Lambert Meertens, Sep 11 2024

A384876 Smallest number m such that both m-1 and m+1 are products of at least n (not necessarily distinct) primes.

Original entry on oeis.org

3, 5, 17, 55, 161, 1457, 2431, 13121, 101249, 153089, 2086399, 7991297, 65071999, 72630271, 2829746177, 2975006719, 68278476799, 75389157377, 159703334911, 1570258288639, 9714181341185, 91845775327231, 551785225781249, 2123044908695551, 4560483868737535, 4560483868737535, 424428773098651649

Offset: 1

Sinuhe Perea, Jun 12 2025

nonn

The sequence is nondecreasing. - David A. Corneth, Jun 13 2025

			The smallest number surrounded by semiprime numbers is 5 (between 4 and 6).
And 17 lies between 16 = 2^4 and 18 = 2*3^2.

Cf. A000040, A001222, A099047, A154704, A176462, A384372.

F:= proc(n) local pq,t,x,y,z,p,i,m;
  uses priqueue;
  initialize(pq);
      insert([-2^n, 2$n], pq);
  y:= -infinity; z:= -infinity;
    do
      t:= extract(pq);
      x:= -t[1];
      if x-y=2 or x-z=2 then return x-1 fi;
      z:= y; y:= x; m:= nops(t);
      if t[-1] = 2 then insert([2*t[1],2$m],pq) fi;
      p:= nextprime(t[-1]);
      for i from m to 2 by -1 while t[i] = t[-1] do
        insert([t[1]*(p/t[-1])^(m+1-i), op(t[2..i-1]), p$(m+1-i)], pq)
      od;
    od
end proc:
seq(F(i),i=1..20); # Robert Israel, Jun 12 2025

a(n) = my(m=2); while((bigomega(m-1)Michel Marcus, Jun 13 2025

a(10)-a(13) from Alois P. Heinz, Jun 12 2025
a(14)-a(20) from Robert Israel, Jun 12 2025
More terms from David A. Corneth, Jun 13 2025

cp's OEIS Frontend

A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A285805 Prime numbers p such that 6*p-1 and 6*p+1 are composite numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A174392 Neither n-1 nor n+1 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A384876 Smallest number m such that both m-1 and m+1 are products of at least n (not necessarily distinct) primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Extensions

A285805 Prime numbers p such that 6p-1 and 6p+1 are composite numbers.