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A101186
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Values of k for which 7m+1, 8m+1 and 11m+1 are prime, with m = 1848k + 942.
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3
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13, 123, 218, 223, 278, 411, 513, 551, 588, 733, 743, 796, 856, 928, 1168, 1226, 1263, 1401, 1533, 1976, 1981, 2013, 2096, 2138, 2241, 2376, 2556, 2676, 2703, 3626, 3703, 3718, 3971, 4008, 4121, 4138, 4163, 4188, 4211, 4313, 4423, 4653, 4656, 4901, 5018
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OFFSET
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1,1
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COMMENTS
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The number (7m+1)(8m+1)(11m+1) is a 3-factor Carmichael number if and only if m is equal to 1848k+942 with k in this sequence. The sequence includes the value k = 10^329 - 4624879 which yields a 1000-digit Carmichael number with three prime factors of 334 digits each. Other Carmichael numbers of the same form would necessarily have 4 prime factors or more; the smallest such example is 3664585=127*(7*29)*199, for m=18.
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LINKS
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EXAMPLE
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a(1)=13 because k=13 corresponds to m=24966, which yields a product of three primes (7m+1)(8m+1)(11m+1) equal to the Carmichael number 9585921133193329. (Among all Carmichael numbers with 16 or fewer digits, as first listed by Richard G. E. Pinch, this one features the largest "least prime factor".)
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MAPLE
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filter:= proc(n) local m;
m:= 1848*n+942;
andmap(isprime, [7*m+1, 8*m+1, 11*m+1])
end proc:
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MATHEMATICA
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q[k_] := Module[{m = 1848*k + 942}, PrimeQ[7*m + 1] && PrimeQ[8*m + 1] && PrimeQ[11*m + 1]]; Select[Range[6000], q] (* Amiram Eldar, Apr 27 2024 *)
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PROG
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(Magma) [k:k in [1..5100]| forall{s:s in [7, 8, 11]|IsPrime(m*s+1) where m is 1848*k+942}]; // Marius A. Burtea, Nov 01 2019
(PARI) is(k) = {my(m = 1848*k + 942); isprime(7*m + 1) && isprime(8*m + 1) && isprime(11*m + 1); } \\ Amiram Eldar, Apr 27 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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