A349321 -id:A349321 - cp's OEIS frontend

A066388 Numbers j such that j and 2j are both between a pair of twin primes.

6, 30, 660, 810, 2130, 2550, 3330, 3390, 5850, 6270, 10530, 33180, 41610, 44130, 53550, 55440, 57330, 63840, 65100, 70380, 70980, 72270, 74100, 74760, 78780, 80670, 81930, 87540, 93240, 102300, 115470, 124770, 133980, 136950, 156420

Offset: 1

Jud McCranie, Dec 23 2001

nonn

Also terms of A014574 such that twice the term is also in A014574. Related to a problem of anti-divisors.

All a(n) > 6 must be a multiple of 30: As for elements of A014574, we must have a(n) = 6k, and k = 5m+-1 would lead to a(n)-+1 divisible by 5, while k = 5m+-2 would lead to 2*a(n)+-1 divisible by 5, so only k=5m is possible. - M. F. Hasler, Nov 27 2010

			j = 30 is a term since 29 and 31 are prime, as are 59 and 61.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Eric Weisstein's World of Mathematics, Bitwin Chain.

Subsequence of A014574.
Subsequences: A118859, A118860, A349321.
Cf. A001359, A006512, A117499.
Cf. A118859, A118860, A349321, A348348.

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; d=2; If[p2-p1==d, w=p1+1; If[PrimeQ[2*w-1]&&PrimeQ[2*w+1], AppendTo[lst, w]]], {n, 1, 10^4}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

{ n=0; forstep (m=2, 10^9, 2, if (isprime(m - 1) && isprime(m + 1) && isprime(2*m - 1) && isprime(2*m + 1), write("b066388.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 13 2010

A117499(a(n)) = 4. - Reinhard Zumkeller, Mar 23 2006

{k in A014574: 2*k in A014574}. - R. J. Mathar, Jan 20 2025

A118860 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1 and 4k+1 are all primes.

Original entry on oeis.org

21968100, 37674210, 81875220, 356467230, 416172330, 750662640, 1007393730, 1150070040, 1586271960, 1963954650, 3127171320, 3669568560, 4377895410, 4383541050, 5575083360, 5686935870, 5708418870, 7365234450, 7478474430, 7681046100, 8453862690, 8898688680

Offset: 1

Labos Elemer, May 03 2006

nonn

All terms are multiples of 210, hence simpler code is possible.

			21968100 is a term because 21968099, 21968101, 43936199, 43936201, 65904299, 65904301, 87872399, 87872401 are all prime.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A014574, A066388 and A118859.
Subsequence: A349321.
Cf. A174293, A348348.

tb={};Do[If[(PrimeQ[n-1]&&PrimeQ[n+1])&& (PrimeQ[2*n-1]&&PrimeQ[2*n+1])&& (PrimeQ[3*n-1]&&PrimeQ[3*n+1])&& (PrimeQ[4*n-1]&&PrimeQ[4*n+1]), Print[n];AppendTo[tb,n]], {n,21968100,10^8,210}];tb
Select[210*Range[424*10^5],AllTrue[{#-1,#+1,2#-1,2#+1,3#-1,3#+1,4#-1,4#+1},PrimeQ]&] (* Harvey P. Dale, Jul 23 2024 *)

isok(k) = if(k % 210, 0, for(i = 1, 4, forstep(j = -1, 1, 2, if(!isprime(i*k-j), return(0)))); 1); \\ Amiram Eldar, Mar 13 2025

a(n) = 210*A174293(n).

Edited by Don Reble, May 16 2006

a(20)-a(22) from Pontus von Brömssen, Oct 14 2021

A118859 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1 and 3k+1 are primes.

Original entry on oeis.org

6, 53550, 420420, 422310, 1624350, 2130240, 3399900, 5199810, 5246010, 6549270, 7384440, 7775880, 9516570, 9565710, 10430280, 11845260, 13207950, 14562870, 14619990, 18747960, 20099940, 21596820, 21968100, 24358950, 24610740, 26916120, 28359240, 30838080

Offset: 1

Labos Elemer, May 03 2006

nonn

			6 is a term because 5, 7, 11, 13, 17, 19 are all prime.

Amiram Eldar, Table of n, a(n) for n = 1..4000

Subsequence of A014574 and A066388.
Subsequences: A118860, A349321.
Cf. A014574, A290811, A348348.

Select[Range[25*10^6],AllTrue[Flatten[{#+{1,-1},2#+{1,-1},3#+{1,-1}}], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 13 2016 *)

isok(k) = isprime(k-1) && isprime(k+1) && isprime(2*k-1) && isprime(2*k+1) && isprime(3*k-1) && isprime(3*k+1); \\ Amiram Eldar, Mar 13 2025

a(n) = 6*A290811(n).

Edited by Don Reble, May 16 2006

a(26)-a(28) from Jon E. Schoenfield, Dec 07 2021

A348348 Smallest k such that the numbers jk - 1 and jk + 1 are prime for 1 <= j <= n.

Original entry on oeis.org

4, 6, 6, 21968100, 100803789240, 683751016938990, 1651735848676253340

Offset: 1

Pontus von Brömssen, Oct 13 2021

The following heuristic argument suggests that a(n) exists for all n: For large (random) k and a specific j <= n, the probability that both j*k - 1 and j*k + 1 are prime should be of the order 1/(log k)^2 (a slight twist of the first Hardy-Littlewood conjecture). Assuming independence between different j, the probability that this holds for 1 <= j <= n is of the order 1/(log k)^(2*n). Since the sum over k of 1/(log k)^(2*n) diverges, this should hold for infinitely many k by the second Borel-Cantelli lemma (assuming independence between different k).

			a(1) = A014574(1) = 4.
a(2) = A066388(1) = 6.
a(3) = A118859(1) = 6.
a(4) = A118860(1) = 21968100.
a(5) = A349321(1) = 100803789240.

Wikipedia, Borel-Cantelli lemma
Wikipedia, Twin prime

Cf. A014574, A066388, A118859, A118860, A349321.

isok(k, n) = for (j=1, n, if (!isprime(j*k-1) || !isprime(j*k+1), return(0))); return(1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jul 01 2022

from sympy import isprime,nextprime
def A348348(n):
    p = 2
    while 1:
        p_next = nextprime(p)
        if p_next == p+2 and all(isprime(j*(p+1)-1) and isprime(j*(p+1)+1) for j in range(2,n+1)):
            return p+1
        p = p_next

a(5), a(6) from Jon E. Schoenfield, Nov 14 2021

a(7) from Klaus Muuss, Jul 01 2022

cp's OEIS Frontend

A066388 Numbers j such that j and 2j are both between a pair of twin primes.

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A118860 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1 and 4k+1 are all primes.

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A118859 Numbers k such that k-1, k+1, 2*k-1, 2*k+1, 3*k-1 and 3*k+1 are primes.

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A348348 Smallest k such that the numbers j*k - 1 and j*k + 1 are prime for 1 <= j <= n.

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A118859 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1 and 3k+1 are primes.

A348348 Smallest k such that the numbers jk - 1 and jk + 1 are prime for 1 <= j <= n.