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%I #9 Apr 21 2024 19:10:37
%S 1,71,155,176,241,346,420,540,690,801,1145,1421,1506,2026,2066,3080,
%T 3235,3371,3445,3511,3640,4746,4925,5681,5901,6055,6520,7931,8365,
%U 8970,9006,9556,9685,10186,11396,11750,11935,12055,12666,13205,13266,13825,13881,14606
%N Numbers m such that 18*m + 1, 36*m + 1, 108*m + 1, and 162*m + 1 are all primes.
%C If m is a term, then (18*m + 1) * (36*m + 1) * (108*m + 1) * (162*m + 1) is a Carmichael number (A002997). These are the Carmichael numbers of the form W_4(3*m) in Nakamula et al. (2007).
%C The corresponding Carmichael numbers are 12490201, 288503529142321, 6548129556412321, ...
%H Amiram Eldar, <a href="/A372188/b372188.txt">Table of n, a(n) for n = 1..10000</a>
%H Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, <a href="https://arxiv.org/abs/math/0702410">New polynomials producing absolute pseudoprimes with any number of prime factors</a>, arXiv:math/0702410 [math.NT], 2007.
%e 1 is a term since 18*1 + 1 = 19, 36*1 + 1 = 37, 108*1 + 1 = 109, and 162*1 + 1 = 163 are all primes.
%e 71 is a term since 18*71 + 1 = 1279, 36*71 + 1 = 2557, 108*71 + 1 = 7669, and 162*71 + 1 = 11503 are all primes.
%t q[n_] := AllTrue[{18, 36, 108, 162}, PrimeQ[#*n + 1] &]; Select[Range[15000], q]
%o (PARI) is(n) = isprime(18*n + 1) && isprime(36*n + 1) && isprime(108*n + 1) && isprime(162*n + 1);
%Y Cf. A002997, A318646, A372190.
%Y Similar sequences: A046025, A257035, A206024, A206349, A372186, A372187.
%K nonn,easy
%O 1,2
%A _Amiram Eldar_, Apr 21 2024