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%I #17 Jun 21 2024 15:25:46
%S 1,8925,70070,70385,270725,355040,566650,866635,874335,1091545,
%T 1230740,1295980,1586095,1594285,1738380,1974210,2201325,2427145,
%U 2436665,3124660,3349990,3599470,3661350,4059825,4101790,4486020,4726540,5139680,5613370,5898655,6279035
%N Numbers n such that (6n-1, 6n+1), (12n-1, 12n+1) and (18n-1, 18n+1) are 3 pairs of twin primes.
%C 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).
%C Intersection of A046025 and A290810.
%C The first 10 pairs of corresponding Lucas-Carmichael and Carmichael numbers ((6n-1)*(12n-1)*(18n-1), (6n+1)*(12n+1)*(18n+1)) are:
%C (935, 1729)
%C (921329139943799, 921392227198801)
%C (445860973748310119, 445864862313790921)
%C (451901165073782759, 451905088679976961)
%C (25715181770344848599, 25715239817629143601)
%C (58001133699332691839, 58001233533626759041)
%C (235803065459494289399, 235803319764534509401)
%C (843555229160685647759, 843555823997214441961)
%C (866240412591524160959, 866241018045184403161)
%C (1685504102154302331719, 1685505045798928055521)
%C (2416038446298343361039, 2416039645957333860241)
%H Tim Johannes Ohrtmann, <a href="/A290811/b290811.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%t seq = {}; Do[ If[ AllTrue[{6 m - 1, 6 m + 1, 12 m - 1, 12 m + 1, 18 m - 1,
%t 18 m + 1}, PrimeQ ], AppendTo[seq, m]], {m, 1, 10^7} ]; seq
%t Select[Range[6280000],AllTrue[{6#+1,6#-1,12#+1,12#-1,18#+1,18#-1},PrimeQ]&] (* _Harvey P. Dale_, Jun 21 2024 *)
%o (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
%Y Cf. A002997, A006972, A033502, A046025, A290810.
%K nonn
%O 1,2
%A _Amiram Eldar_, Aug 11 2017
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