A066388 -id:A066388 - cp's OEIS frontend

A069142 Primes p such that p+2, 2p+1, and 2p+3 are also prime.

5, 29, 659, 809, 2129, 2549, 3329, 3389, 5849, 6269, 10529, 33179, 41609, 44129, 53549, 55439, 57329, 63839, 65099, 70379, 70979, 72269, 74099, 74759, 78779, 80669, 81929, 87539, 93239, 102299, 115469, 124769, 133979, 136949, 156419

Offset: 1

Neil Fernandez, Apr 08 2002

Previous name: Lower prime in a twin pair that yields another.

a(n) gives the terms for A005382(i)-A005384(j)=2. - J. M. Bergot, Mar 12 2015

			659 and 661 form a prime twin pair. Their sum is 1320. 1320 is sandwiched between 1319 and 1321, which form another prime twin pair. So 659 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A014574.
Cf. A005384, A005382.
Cf. A066388.

[p: p in PrimesUpTo(160000) | IsPrime(p+2) and IsPrime(2*p+1) and IsPrime(2*p+3)]; // Vincenzo Librandi, Apr 09 2013

p = q = 1; Do[q = Prime[n]; If[p + 2 == q && PrimeQ[2p + 1] && PrimeQ[2p + 3], Print[p]]; p = q, {n, 1, 10^4}]
Select[Prime[Range[15000]], PrimeQ[# + 2] && PrimeQ[2 # + 1] && PrimeQ[2 # + 3]&] (* Vincenzo Librandi, Apr 09 2013 *)

forprime(p=1,10^5,if(isprime(p+2)&&isprime(2*p+1)&&isprime(2*p+3),print1(p,", "))) \\ Derek Orr, Mar 11 2015

a(n) = A066388(n)-1. - R. J. Mathar, Nov 02 2023

Edited and extended by Robert G. Wilson v, Apr 11 2002

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

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

A117499 Number of subsets of {n-1, n, n+1} that sum up to a prime.

Original entry on oeis.org

4, 4, 4, 3, 2, 4, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 0, 1, 1, 1, 2, 4, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 0, 2, 2, 1, 1, 2, 2, 2, 0, 1, 1, 1, 2, 3, 1, 0, 0, 0, 1, 3, 2, 2, 2, 2, 1, 2, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Mar 23 2006

nonn

0 <= a(n) <= 4; a(A066388(n)) = 4.

a(A221309(n)) = 0; a(A221310(n)) = 4. - Reinhard Zumkeller, Jan 10 2013

			a(1) = #{2, 0+2=2, 1+2=3, 0+1+2=3} = 4;
a(2) = #{2, 3, 1+2=3, 2+3=5} = 4;
a(3) = #{2, 3, 2+3=5, 3+4=7} = 4;
a(4) = #{3, 5, 3+4=7} = 3;
a(5) = #{5, 5+6=11} = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051.

a117499 1 = sum $ map a010051 [1, 2, 0 + 1, 0 + 2, 1 + 2, 0 + 1 + 2]
a117499 n = sum $ map a010051 [n - 1, n, n + 1, 2 * n - 1, 2 * n + 1]
-- Reinhard Zumkeller, Jan 10 2013

Table[Length[Select[{-1+n,n,1+n,-1+2 n,2 n,1+2 n,3 n},PrimeQ]],{n,105}]
ssp[{a_,b_,c_}]:=Count[Subsets[{a,b,c},3],?(PrimeQ[Total[#]]&)]; ssp/@ Partition[ Range[0,110],3,1] (* _Harvey P. Dale, Jan 29 2013 *)

a(n) = A010051(n-1) + A010051(n) + A010051(n+1) + A010051(2*n-1) + A010051(2*n) + A010051(2*n+1).

A185145 Smallest average of twin prime pairs s such that n*s is also average of twin prime pairs.

Original entry on oeis.org

4, 6, 4, 18, 6, 12, 6, 30, 12, 6, 18, 6, 150, 30, 4, 12, 6, 4, 12, 12, 42, 30, 6, 18, 6, 12, 4, 270, 12, 6, 42, 6, 6, 30, 12, 12, 180, 6, 60, 6, 30, 150, 30, 30, 4, 18, 6, 4, 18, 12, 42, 6, 150, 30, 12, 60, 4, 6, 18, 4, 462, 180, 1230, 18, 30, 108, 60, 180, 12

Offset: 1

Manuel Valdivia, Mar 12 2012

nonn

Probably for all n>1 and also for all average s there are at least an average n*s. Note that this is equivalent to the Twin Prime Conjecture. Verified n to 10^7. First consecutive averages: 4 to 34260.

			A014574(12) = 150, then 13*150 = 1950 = A014574(60).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A066388, A014574.

t=Select[Table[Prime[n] + 1, {n, 10^4}], PrimeQ[#1 + 1] & ]; Table[s:=t[[m]]; m=1; While[!PrimeQ[n*s-1] || !PrimeQ[n*s+1], m++]; s, {n,1,100}]

a(n) = A014574(j) if n*A014574(j) = A014574(k).

A069175 Numbers k such that k-1, k+1, 2k-1, 2k+1, 4k-1 and 4k+1 are all prime.

Original entry on oeis.org

211050, 248640, 253680, 410340, 507360, 605640, 1121190, 1138830, 1262100, 2162580, 2172870, 2277660, 4070220, 6305460, 7671510, 11659410, 12577110, 14203770, 14862120, 17472840, 18728640, 18798360, 20520570, 21140700

Offset: 1

Don Reble, Apr 09 2002

nonn

			211050 is in the sequence because 211049, 211051, 422099, 422101, 844199 and 844201 are all prime.

Cf. A066388.

isA069175 := proc(k)
    if isprime(k-1) and isprime(k+1) and isprime(2*k-1) and isprime(2*k+1) and isprime(4*k-1) and isprime(4*k+1) then
        true ;
    else
        false;
    end if;
end proc:
n := 1 :
for k from 4 by 2 do # create b-file
    if isA069175(k) then
        printf("%d %d\n",n,k) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Nov 02 2023

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[2*n-1]&&PrimeQ[2*n+1]&&PrimeQ[4*n-1]&&PrimeQ[4*n+1],Print[n];AppendTo[lst,n]],{n,11!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 05 2009 *)
Select[Range[21150000],AllTrue[{#-1,#+1,2#-1,2#+1,4#-1,4#+1},PrimeQ]&] (* Harvey P. Dale, Aug 19 2025 *)

Offset changed to 1 by Georg Fischer, Sep 23 2022

A076504 Numbers k such that (k-1, k+1) and (k/2-1, k/2+1) are both pairs of twin primes.

Original entry on oeis.org

12, 60, 1320, 1620, 4260, 5100, 6660, 6780, 11700, 12540, 21060, 66360, 83220, 88260, 107100, 110880, 114660, 127680, 130200, 140760, 141960, 144540, 148200, 149520, 157560, 161340, 163860, 175080, 186480, 204600, 230940, 249540

Offset: 1

Eric W. Weisstein, Oct 15 2002

nonn

Terms after the first are multiples of 60. - Marc Morgenegg, Apr 19 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A066388.

Select[Mean /@ Select[Partition[Prime[Range[30000]], 2, 1], Last[#] - First[#] == 2 &], And @@ PrimeQ[#/2 + {1, -1}] &] (* Harvey P. Dale, Jan 23 2014 *)

a(n) = 2*A066388(n).

Definition corrected by Jaroslav Krizek, Apr 12 2009

A363500 Numbers k between twin primes p, q where k+p and k+q are also twin primes, and kp and kq are between twin primes.

Original entry on oeis.org

6, 109505970, 1519435260, 22606027290, 25980888360, 33995114580, 42029719620, 45284475810, 56527358160, 63402770550, 73924546080, 82625597670, 121883654550, 150444654360, 192416460810, 210205659510, 258719413680, 270709718160, 284455564050, 309050171430

Offset: 1

Bryce Case, Jr. and Antonio Gimenez, Jun 05 2023

nonn

Larger twin primes are found on either side of 6x, so my highly-unoptimized code simply keeps adding 6 and performing the requisite primality checks using golang's "ProbablyPrime" function, a combination of Miller-Rabin and Baillie-PSW, accurate up to 2^64. Based on seminal work by fellow OEIS contributor Antonio Gimenez.
To generate, k = 6x.
p = k-1, q = k+1, check the primality of k+p, k+q, then check the primality of ((k*p) +/- 1) and ((k*q) +/- 1).
If k > x+1 and x > 1, then all eight primes are not divisible by x. If k > 8, then k == 0 (mod 210). - Jason Yuen, Jun 02 2024

Jason Yuen, Table of n, a(n) for n = 1..10000 (first 49 terms from Bryce Case, Jr.)
Bryce Case, Jr., a363500.go

Subsequence of A066388.

Cf. A364263.

Go
```
// See link.
```

a(n) = 210*A364263(n-1) for n > 1. - Hugo Pfoertner, Jun 03 2024

cp's OEIS Frontend

A069142 Primes p such that p+2, 2p+1, and 2p+3 are also prime.

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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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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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A117499 Number of subsets of {n-1, n, n+1} that sum up to a prime.

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A185145 Smallest average of twin prime pairs s such that n*s is also average of twin prime pairs.

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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.

A069175 Numbers k such that k-1, k+1, 2k-1, 2k+1, 4k-1 and 4k+1 are all prime.

A363500 Numbers k between twin primes p, q where k+p and k+q are also twin primes, and kp and kq are between twin primes.