A182481 -id:A182481 - cp's OEIS frontend

A182482 6nA182481(n)-1.

5, 11, 17, 71, 29, 71, 41, 191, 107, 59, 197, 71, 311, 419, 179, 191, 101, 107, 227, 239, 881, 659, 137, 431, 149, 311, 809, 2687, 347, 179, 1301, 191, 197, 1019, 419, 431, 1997, 227, 1871, 239, 1229, 2267, 1031, 1319, 269, 827, 281, 1151, 881, 599

Offset: 1

Vladimir Shevelev, May 01 2012

nonn

By the construction of A182481, every term is lesser of twin primes.
Every lesser more than 3 of twin primes appears in the sequence.
Number m=(a(n)+1)/6 is the place of the last appearance of a(n); m is multiple of all previous places of the appearance of a(n), if they exist.
In particular, a(n) appears only once, if (a(n)+1)/6 is 1 or prime (in this case n is 1 or prime and A182481(n)=1). Conversely is not true. For example, a(10)=59 appears only once, although 10 is not prime.

			All places where 71 appears are 4,6,12. "Thus" 12 is multiple of 4 and 6.
Since (101+1)/6=17 is prime, then 101 appears only once.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A001359, A006512, A014574, A001097, A077800, A002822.

A182521 Numbers n such that A182481(n)=1 and there is not a representation n=d_1*d_2 with d_2>1, such that A182481(d_1)=d_2.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 17, 23, 25, 45, 47, 77, 87, 95, 103, 107, 137, 143, 175, 215, 247, 283, 287, 313, 347, 355, 373, 385, 397, 425, 443, 455, 467, 565, 577, 593, 637, 653, 667, 703, 737, 773, 775, 787, 850, 907, 913, 917, 943, 975, 1033, 1075, 1117, 1127, 1130

Offset: 1

Vladimir Shevelev, May 03 2012

nonn

Or the numbers n such that 6*n-1 is lesser of twin primes which occurs in A182482 only once.

All terms of A060212 are in the sequence.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A182482, A182483, A182513, A060212.

Numbers n for which A182483(A182513(n))/n = A182481(n) = 1.

Insert 1 and more terms from Ray Chandler, Sep 18 2019

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

Original entry on oeis.org

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

A294731 Smallest average of a twin prime pair divisible by the n-th prime, i.e. A090530(n), divided by 6*prime(n).

Original entry on oeis.org

1, 1, 3, 4, 1, 2, 1, 2, 7, 9, 5, 4, 1, 20, 3, 43, 4, 3, 14, 22, 9, 8, 19, 7, 1, 1, 8, 4, 24, 5, 1, 2, 2, 13, 4, 6, 5, 9, 22, 3, 15, 6, 11, 3, 7, 5, 20, 5, 6, 7, 3, 3, 9, 14, 10, 2, 35, 2, 1, 10, 25, 17, 1, 35, 5, 4, 1, 18, 15, 12, 25, 1, 2, 5

Offset: 3

Hugo Pfoertner, Nov 09 2017

The sequence starts at n=3, because A090530(1)=4 is not divisible by 6*2 and A090530(2)=6 is not divisible by 6*3.

The positions of ones in the sequence are given by A060212, i.e. a(A000720(A060212(n)))=1 for all n>=3.

			a(5)=3 because 198 is the smallest average of a twin prime pair {197,199} that is divisible by the 5th prime 11: 3 = 198 / (6*11).

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A014574, A060212, A071407, A090530, A182481.

a[n_] := Module[{p = Prime[n], k = 1}, While[! PrimeQ[6*k*p - 1] || ! PrimeQ[6*k*p + 1], k++]; k]; Array[a, 100, 3] (* Amiram Eldar, Aug 25 2025 *)

a(n) = A090530(n) / ( 6 * prime(n) ) for n >= 3.

a(n) = A071407(n) / 6. - Amiram Eldar, Aug 25 2025

A182483 a(n) is the least m such that A182482(m) = A001359(n), the n-th twin prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 4, 17, 9, 23, 25, 15, 8, 11, 19, 20, 45, 47, 13, 29, 14, 24, 77, 87, 95, 50, 103, 107, 22, 27, 137, 46, 143, 21, 34, 43, 175, 59, 91, 48, 41, 71, 215, 31, 44, 119, 121, 247, 62, 67, 54, 139, 283, 287, 149, 39, 313, 161, 65, 37, 169, 347, 116

Offset: 2

Vladimir Shevelev, May 01 2012

nonn

a(n) exists for every n>=2.

Ray Chandler, Table of n, a(n) for n = 2..10001

Cf. A182481, A182482, A001359, A006512, A014574, A001097, A077800, A002822.

t = Table[k = 0; While[p = 6*k*n - 1; ! (PrimeQ[p] && PrimeQ[p + 2]), k++]; p, {n, 1000}]; tp = Select[Prime[Range[1000]], PrimeQ[# + 2] &]; t2 = {}; found = True; n = 2; While[found, pos = Position[t, tp[[n]], 1, 1]; If[pos == {}, found = False, AppendTo[t2, pos[[1, 1]]]; n++]]; t2 (* T. D. Noe, May 02 2012 *)

A182513 a(n) is the solution of equation A001359(x) = A182482(n).

Original entry on oeis.org

2, 3, 4, 8, 5, 8, 6, 14, 10, 7, 15, 8, 20, 22, 13, 14, 9, 10, 16, 17, 35, 30, 11, 23, 12, 20, 31, 76, 21, 13, 45, 14, 15, 36, 22, 23, 61, 16, 57, 17, 42, 69, 37, 46, 18, 33, 19, 41, 35, 27, 67, 20, 149, 52, 30, 76, 123, 21, 39, 171, 282, 50, 69, 41, 60, 84, 51, 98, 33, 22, 43

Offset: 1

Vladimir Shevelev, May 03 2012

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A182482, A182483, A001359.

More terms from Ray Chandler, Sep 18 2019

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

cp's OEIS Frontend

A182482 6*n*A182481(n)-1.

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A182521 Numbers n such that A182481(n)=1 and there is not a representation n=d_1*d_2 with d_2>1, such that A182481(d_1)=d_2.

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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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A294731 Smallest average of a twin prime pair divisible by the n-th prime, i.e. A090530(n), divided by 6*prime(n).

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A182483 a(n) is the least m such that A182482(m) = A001359(n), the n-th twin prime.

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A182513 a(n) is the solution of equation A001359(x) = A182482(n).

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

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A182482 6nA182481(n)-1.

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