A290811 Numbers n such that (6n-1, 6n+1), (12n-1, 12n+1) and (18n-1, 18n+1) are 3 pairs of twin primes.

1, 8925, 70070, 70385, 270725, 355040, 566650, 866635, 874335, 1091545, 1230740, 1295980, 1586095, 1594285, 1738380, 1974210, 2201325, 2427145, 2436665, 3124660, 3349990, 3599470, 3661350, 4059825, 4101790, 4486020, 4726540, 5139680, 5613370, 5898655, 6279035

Offset: 1

Author

Amiram Eldar, Aug 11 2017

Keywords

• nonn

Comments

If n is in the sequence then (6n+1)*(12n+1)*(18n+1) is a Carmichael number (A002997) and (6n-1)*(12n-1)*(18n-1) is a Lucas-Carmichael number (A006972).

Intersection of A046025 and A290810.

The first 10 pairs of corresponding Lucas-Carmichael and Carmichael numbers ((6n-1)*(12n-1)*(18n-1), (6n+1)*(12n+1)*(18n+1)) are:

(935, 1729)

(921329139943799, 921392227198801)

(445860973748310119, 445864862313790921)

(451901165073782759, 451905088679976961)

(25715181770344848599, 25715239817629143601)

(58001133699332691839, 58001233533626759041)

(235803065459494289399, 235803319764534509401)

(843555229160685647759, 843555823997214441961)

(866240412591524160959, 866241018045184403161)

(1685504102154302331719, 1685505045798928055521)

(2416038446298343361039, 2416039645957333860241)

Examples

			1 is in the sequence since (6*1 - 1, 6*1 + 1) = (5, 7), (12*1 - 1, 12*1 + 1) = (11, 13) and (18*1 - 1, 18*1 + 1) = (17, 19) are all pairs of twin primes.
		

Links

• Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002997, A006972, A033502, A046025, A290810.

Programs

Mathematica

seq = {}; Do[ If[ AllTrue[{6 m - 1, 6 m + 1, 12 m - 1, 12 m + 1, 18 m - 1,
    18 m + 1}, PrimeQ ], AppendTo[seq, m]], {m, 1, 10^7} ]; seq
Select[Range[6280000],AllTrue[{6#+1,6#-1,12#+1,12#-1,18#+1,18#-1},PrimeQ]&] (* Harvey P. Dale, Jun 21 2024 *)

PARI

isok(n) = isprime(6*n-1) && isprime(6*n+1) && isprime(12*n-1) && isprime(12*n+1) && isprime(18*n-1) && isprime(18*n+1); \\ Michel Marcus, Aug 11 2017