A294731 -id:A294731 - cp's OEIS frontend

A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.

2, 3, 5, 7, 17, 23, 47, 103, 107, 137, 283, 313, 347, 373, 397, 443, 467, 577, 593, 653, 773, 787, 907, 1033, 1117, 1423, 1433, 1613, 1823, 2027, 2063, 2137, 2153, 2203, 2287, 2293, 2333, 2347, 2677, 2903, 3257, 3307, 3407, 3413, 3593, 3623, 3673, 3923

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Primes in A182521. Also all primes p for which A182481(p)=1. - Vladimir Shevelev, May 03 2012

Conjecture: a(n) ~ n*log(n)*log(n*log(n))*log(log(n)). - Carl R. White, Nov 16 2023

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A001359, A002822, A014574, A027856, A058383, A059960, A182481, A182521, A294731.

lst={}; Do[p=Prime[n]; If[PrimeQ[6*p-1] && PrimeQ[6*p+1], AppendTo[lst,p]], {n,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)

forprime(p=2, 9999, if(isprime(6*p+1) & isprime(6*p-1), print(p))) \\ David Radcliffe, Apr 02 2016

from sympy import *; print([p for p in primerange(2,9999) if isprime(6*p-1) and isprime(6*p+1)]) # David Radcliffe, Apr 02 2016

A071407 Least k such that kprime(n) + 1 and kprime(n) - 1 are twin primes.

Original entry on oeis.org

2, 2, 6, 6, 18, 24, 6, 12, 6, 12, 42, 54, 30, 24, 6, 120, 18, 258, 24, 18, 84, 132, 54, 48, 114, 42, 6, 6, 48, 24, 144, 30, 6, 12, 12, 78, 24, 36, 30, 54, 132, 18, 90, 36, 66, 18, 42, 30, 120, 30, 36, 42, 18, 18, 54, 84, 60, 12, 210, 12, 6, 60, 150, 102, 6, 210, 30, 24, 6

Offset: 1

Labos Elemer, May 24 2002

Note that 6 divides a(n) for n > 2. - T. D. Noe, Jan 07 2013

			n=4: prime(4)=7, a(4)=6 because 6*prime(4)=42 and {41,43} are primes.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A071256, A071404-A071406, A060256, A060210, A090530, A294731.
Cf. A071558 (k at every integer).
Cf. A220141, A220142 (record values).

a071407 n = head [k | k <- [2,4..], let x = k * a000040 n,
                      a010051' (x - 1) == 1, a010051' (x + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

Table[fl=1; Do[s=(Prime[j])*k; If[PrimeQ[s-1]&&PrimeQ[s+1]&&Equal[fl, 1], Print[{j, k}]; fl=0], {k, 1, 2*j^2}], {j, 0, 100}]

From Amiram Eldar, Aug 25 2025: (Start)
a(n) = A090530(n) / prime(n).
a(n) = 6 * A294731(n) for n >= 3. (End)

A090530 Least multiple k of prime(n) such that (k-1,k+1) forms a twin prime pair, or 0 if no such number exists.

Original entry on oeis.org

4, 6, 30, 42, 198, 312, 102, 228, 138, 348, 1302, 1998, 1230, 1032, 282, 6360, 1062, 15738, 1608, 1278, 6132, 10428, 4482, 4272, 11058, 4242, 618, 642, 5232, 2712, 18288, 3930, 822, 1668, 1788, 11778, 3768, 5868, 5010, 9342, 23628, 3258, 17190

Offset: 1

Amarnath Murthy, Dec 07 2003

a(n) is a multiple of 6*prime(n) for n>2. Conjecture: No term is zero.

			a(5) = 198 = 11*18, (197,199) forms a twin prime pair.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A014574, A071407, A090531, A294731 [a(n)/(6*prime(n))].

For[n = 1, n < 40, n++, a := Prime[n]; k := 2; While[Not[PrimeQ[k*a + 1] && PrimeQ[k*a - 1]], k += 2]; Print[k*a]] (* Stefan Steinerberger, Feb 17 2006 *)

a(n) = { my(k = 2, p = prime(n)); while (! (isprime(k*p-1) && isprime(k*p+1)), k++); k*p;} \\ Michel Marcus, Nov 12 2017

a(n) = prime(n) * A071407(n). - Amiram Eldar, Aug 25 2025

More terms from David Wasserman, Dec 21 2005

More terms from Stefan Steinerberger, Feb 17 2006

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

A387287 Primes in the order of their first appearance among the factors of the averages of twin prime pairs.

Original entry on oeis.org

2, 3, 5, 7, 17, 23, 11, 19, 47, 13, 29, 103, 107, 137, 43, 59, 41, 71, 31, 67, 139, 283, 149, 313, 37, 347, 373, 397, 443, 113, 467, 271, 181, 281, 577, 593, 199, 157, 653, 131, 101, 89, 241, 83, 251, 379, 773, 787, 167, 109, 907, 163, 73, 1033, 53, 223, 1117

Offset: 1

Tamas Sandor Nagy, Aug 25 2025

Will every prime appear, so that this sequence is a permutation of the primes?

The answer is yes if A071256(n) exists for every n. - Robert Israel, Aug 25 2025

			a(1) = 2 because 2 appeared first as a prime factor of the average of a twin prime pair, namely of 4 = 2*2 = 2^2, the average of 3 and 5, the first twin prime pair.
a(2) = 3 because 3 appeared next as a prime factor of the average of a twin prime pair, here 6 = 2*3, of the twin primes 5 and 7.
a(3) = 5 because 5 appeared next as a prime factor of the average of a twin prime pair, this time of 30 = 2*3*5, between 29 and 30. The averages 12 and 18 are skipped as their factors, 2 and 3, already appeared.
a(5) = 17 following a(4) = 7, skipping the primes 11 and 13 in the order of appearances.

Cf. A000040, A014574, A071256, A071407, A090530, A294731.

P:= select(isprime, {seq(i,i=3..10^4,2)}):
TPA:= map(`+`, P intersect map(`-`,P,2),1):
TPA:= sort(convert(TPA,list)):
R:= NULL: S:= {}:
for t in TPA do
  V:= numtheory:-factorset(t) minus S;
  if nops(V) > 1 then printf("t = %d: %a\n",t,V) fi;
  R:= R, op(sort(convert(V,list)));
  S:= S union V;
od:
R; # Robert Israel, Aug 25 2025

With[{m = Select[Prime[Range[1000]], PrimeQ[# + 2] &] + 1}, DeleteDuplicates[Flatten[FactorInteger[#][[;; , 1]] & /@ m]]] (* Amiram Eldar, Aug 25 2025 *)

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A060212 Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.

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A071407 Least k such that k*prime(n) + 1 and k*prime(n) - 1 are twin primes.

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A090530 Least multiple k of prime(n) such that (k-1,k+1) forms a twin prime pair, or 0 if no such number exists.

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

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A387287 Primes in the order of their first appearance among the factors of the averages of twin prime pairs.

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A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.

A071407 Least k such that kprime(n) + 1 and kprime(n) - 1 are twin primes.