A060212 -id:A060212 - cp's OEIS frontend

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

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", ")))}

A358381 Primes p such that q1=6p-1 and q2=6p+1 are also primes (twin primes) and q1 is a Sophie Germain prime (i.e., 2*q1+1 is prime).

Original entry on oeis.org

2, 5, 7, 47, 107, 907, 2137, 2347, 3407, 4547, 4597, 8377, 9067, 9277, 9767, 14537, 16427, 18307, 19507, 19997, 23447, 23917, 26927, 27437, 28837, 29297, 33037, 37307, 38327, 45127, 46457, 50957, 52957, 55897, 59077, 59407, 60317, 63667, 65497, 69767, 74377, 77527, 86587, 86837

Offset: 1

Tamas Nagy, Nov 12 2022

nonn

Except for the first 2 terms, every term's last digit is a 7 in base 10.

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

Subsequence of A060212.

Cf. A005384.

filter:= p -> andmap(isprime, [p, 6*p-1, 6*p+1, 12*p-1]):
select(filter, [2,5,seq(i,i=7..10^5,10)]); # Robert Israel, Dec 23 2022

Select[Prime[Range[8500]], PrimeQ[6*# - 1] && PrimeQ[6*# + 1] && PrimeQ[12*# - 1] &] (* Amiram Eldar, Nov 13 2022 *)

A383475 Numbers k such that k*2^d is the average of a twin prime pair for some divisor d of k.

Original entry on oeis.org

2, 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 39, 42, 45, 48, 51, 54, 60, 69, 72, 75, 78, 81, 90, 96, 99, 105, 108, 114, 120, 129, 132, 135, 141, 144, 150, 156, 165, 168, 174, 180, 186, 192, 201, 210, 216, 228, 231, 234, 240, 252, 258, 261, 264, 270, 282, 285, 288, 300

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2025

nonn

			2 is a term in the sequence because 2*2^1 = 4 is the average of twin primes 3 and 5 for divisor d = 1 of k = 2.

Supersequence of 3*A002822 and 3*A060212.

Cf. A014574, A027750, A382811, A383473.

[k: k in [1..300] | not #[d: d in Divisors(k) | IsPrime(k*2^d-1) and IsPrime(k*2^d+1)] eq 0];

q[k_] := AnyTrue[Divisors[k], And @@ PrimeQ[k * 2^# + {-1, 1}] &]; Select[Range[300], q] (* Amiram Eldar, Apr 28 2025 *)

isok(k) = fordiv(k, d, if (isprime(k*2^d-1) && isprime(k*2^d+1), return(1))); return(0); \\ Michel Marcus, Apr 28 2025

A386724 Twin primes p such that 6p+1, 6p-1 is a twin prime pair.

Original entry on oeis.org

3, 5, 7, 17, 103, 107, 137, 283, 313, 347, 1033, 2027, 3257, 3673, 4217, 4547, 5023, 9433, 9767, 11833, 14593, 15137, 15733, 18253, 19423, 20717, 20983, 23537, 25847, 26113, 28753, 32057, 32323, 33073, 35053, 37307, 38327, 39163, 43607, 44623, 46183, 46273, 47743, 48407

Offset: 1

Marc Morgenegg, Jul 31 2025

nonn

{3,5} and {5,7} are the only twin prime pairs occurring in this since (6p-1)*(6p+1)*(6p+11)*(6p+13) is always divisible by 5. Therefore the smallest possible gaps for p>7 is 4 (cousin primes).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002822, A001359, A014574, A176131 (subsequence), A182481, A294731. Subset of A060212.

q:= p-> isprime(p) and ormap(isprime, [p-2, p+2]) and andmap(isprime, [6*p-1, 6*p+1]):
select(q, [2*i+1$i=1..25000])[];  # Alois P. Heinz, Jul 31 2025

Select[Prime[Range[5000]], Or @@ PrimeQ[# + {-2, 2}] && And @@ PrimeQ[6*# + {-1, 1}] &] (* Amiram Eldar, Jul 31 2025 *)

More terms from Pontus von Brömssen, Jul 31 2025

A285983 Prime numbers p such that 3*p has distance <= 2 from the nearest twin prime number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 37, 47, 59, 61, 67, 79, 89, 103, 139, 173, 191, 199, 269, 271, 277, 293, 349, 353, 383, 409, 431, 433, 439, 541, 557, 643, 677, 709, 757, 769, 863, 887, 911, 929, 991, 1039, 1087, 1109, 1123, 1129, 1153, 1181, 1187

Offset: 1

Dimitris Valianatos, Apr 29 2017

nonn

Also prime numbers distance <= 1 from an element of A167379. - Danny Rorabaugh, May 04 2017

Cf. A001359, A006512, A060212.

fQ[n_] := (PrimeQ[3n -4] && PrimeQ[3n -2]) || (PrimeQ[3n +2] && PrimeQ[3n +4]); Join[{2}, Select[ Prime@ Range@ 200, fQ]] (* Robert G. Wilson v, Apr 30 2017 *)

{
print1(2", ");
forprime(n=3,1000,
         p3=3*n;
         if((isprime(p3+2)&&isprime(p3+4))||(isprime(p3-2)&&isprime(p3-4)),
             print1(n", ")
           )
        )
}

cp's OEIS Frontend

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

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A358381 Primes p such that q1=6*p-1 and q2=6*p+1 are also primes (twin primes) and q1 is a Sophie Germain prime (i.e., 2*q1+1 is prime).

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A383475 Numbers k such that k*2^d is the average of a twin prime pair for some divisor d of k.

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A386724 Twin primes p such that 6p+1, 6p-1 is a twin prime pair.

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A285983 Prime numbers p such that 3*p has distance <= 2 from the nearest twin prime number.

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

A358381 Primes p such that q1=6p-1 and q2=6p+1 are also primes (twin primes) and q1 is a Sophie Germain prime (i.e., 2*q1+1 is prime).