A118860 -id:A118860 - cp's OEIS frontend

A174293 A118860/210.

104610, 179401, 389882, 1697463, 1981773, 3574584, 4797113, 5476524, 7553676, 9352165, 14891292, 17474136, 20847121, 20874005, 26548016, 27080647, 27182947, 35072545, 35611783, 36576410, 40256489, 42374708, 51527038, 53528510, 55871968, 56270421, 57824688

Offset: 1

Zak Seidov, Nov 27 2010

nonn

Are all terms composite?

Cf. A118860 Numbers n such that n-/+1, 2n-/+1, 3n-/+1, 4n-/+1 are all primes.

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

Original entry on oeis.org

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

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

A349321 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1, 4k+1, 5k-1, and 5k+1 are all primes.

Original entry on oeis.org

100803789240, 441913177860, 1768738337520, 3906037699410, 5988326187690, 6266477200830, 6905441609220, 6973884137220, 14323608903450, 17683172090430, 20047266723330, 23371434572640, 27904703386560, 29484744885750, 31141493827290, 33202639844220, 34645262968470

Offset: 1

Jon E. Schoenfield, Nov 14 2021

nonn

All terms are multiples of 2*3*5*7*11 = 2310.
From Jon E. Schoenfield, Mar 21 2022: (Start)
Each term is congruent to one of only
       1 residue  modulo 2*3*5*7*11     =         2310 (0.04329%),
       3 residues modulo 2*3*5*7*11*13  =        30030 (0.00999%),
      21 residues modulo 2*3*5*7*...*17 =       510510 (0.00411%),
     189 residues modulo 2*3*5*7*...*19 =      9699690 (0.00195%),
    2457 residues modulo 2*3*5*7*...*23 =    223092870 (0.00110%),
   46683 residues modulo 2*3*5*7*...*29 =   6469693230 (0.00072%),
  980343 residues modulo 2*3*5*7*...*31 = 200560490130 (0.00049%), etc.;
making use of these can allow more efficient searching for terms of the sequence.
The Magma program (see Links) generates a list of the possible residues modulo 2*3*5*7*...*31 and tests only numbers having one of those residues. (Note that the program, when run on the Online Magma Calculator, generates only the first three terms of the sequence before being terminated on reaching the 120-second time limit.) (End)

Jon E. Schoenfield, Magma program

Cf. A014574, A066388, A118859, A118860, A348348.

is_ok(k)=for(j=1,5, if(!isprime(j*k-1), return(0)); if(!isprime(j*k+1), return(0));); return(1); \\ Joerg Arndt, Nov 15 2021

A069176 Numbers n such that n-1, n+1, 2n-1, 2n+1, 4n-1, 4n+1, 8n-1 and 8n+1 are all prime.

Original entry on oeis.org

253680, 1138830, 58680930, 90895770, 124253010, 269877300, 392071680, 613813200, 1014342210, 1277981670, 1413015030, 1453978680, 1753585680, 2919331380, 3424037190, 3538972710, 4025789040, 4175762010, 4362439200

Offset: 1

Don Reble, Apr 09 2002

nonn

For one such number, 41887255410, even 16n-1 and 16n+1 are primes.

			253680 is there because 253679, 253681, 507359, 507361, 1014719, 1014721, 2029439 and 2029441 are all prime.

Cf. A118860. Subset of A069175.

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A174293 A118860/210.

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A066388 Numbers j such that j and 2j are both between a pair of twin 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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A349321 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1, 4k+1, 5k-1, and 5k+1 are all primes.

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A069176 Numbers n such that n-1, n+1, 2n-1, 2n+1, 4n-1, 4n+1, 8n-1 and 8n+1 are all prime.

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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.