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A364003 Integers K such that PSL_2(K) is a K_4-simple group, i.e., |PSL_2(K)| has 4 distinct prime divisors.

%I A364003 #37 Jun 30 2024 22:40:09
%S A364003 11,13,16,19,23,25,27,31,32,37,47,49,53,73,81,97,107,127,128,163,193,
%T A364003 243,257,383,487,577,863,1153,2187,2593,2917,4373,8192,8747,131072,
%U A364003 524288,995327,1492993,1594323,1990657,5308417,28311553,86093443,2147483648,6879707137
%N A364003 Integers K such that PSL_2(K) is a K_4-simple group, i.e., |PSL_2(K)| has 4 distinct prime divisors.
%C A364003 This sequence is conjectured to be infinite, see Bugeaud, Cao, & Mignotte.
%H A364003 Charles R Greathouse IV, <a href="/A364003/b364003.txt">Table of n, a(n) for n = 1..81</a>
%H A364003 Y. Bugeaud, Z. Cao, and M. Mignotte, <a href="https://doi.org/10.1006/jabr.2000.8742">On Simple K4-Groups</a>, Journal of Algebra, 241 (2001), 658-668.
%e A364003 |PSL_2(11)| = 660 = 2^2 * 3 * 5 * 11.
%o A364003 (GAP) is:=function(n)
%o A364003 return IsPrimePowerInt(n) and Length(Unique(FactorsInt(n*(n^2-1))))=4;
%o A364003 end;
%o A364003 Filtered([2..1000], n -> is(n)); # _Charles R Greathouse IV_, Jul 03 2023; edited by _Lixin Zheng_, Jun 23 2024
%o A364003 (PARI) is(n)=isprimepower(n) && omega(lcm([n-1,n,n+1]))==4 \\ _Charles R Greathouse IV_, Jul 03 2023
%o A364003 (PARI) H(n)=isprimepower(n/2^valuation(n,2)/3^valuation(n,3))
%o A364003 list(lim)=my(v=List(), N); lim\=1; for(n=1, logint(lim\2+1, 3), N=2*3^n; while(N<=lim+1, if(isprimepower(N-1) && H(N-2), listput(v, N-1)); if(isprimepower(N+1) && H(N+2) && N+1<=lim, listput(v, N+1)); N<<=1)); for(n=4,logint(N+1,2), N=2^n; if(H(N-1) && H(N+1) && N<=lim, listput(v,N)); if(isprimepower(N-1) && H(N-2), listput(v,N-1)); if(isprimepower(N+1) && H(N+2) && N+1<=lim, listput(v,N+1))); for(n=3,logint(N,3), N=3^n; if(H(N-1) && H(N+1), listput(v,N))); Set(v) \\ _Charles R Greathouse IV_, Jul 03 2023
%Y A364003 Subsequence of A000961.
%Y A364003 Cf. A003586.
%K A364003 nonn
%O A364003 1,1
%A A364003 _Lixin Zheng_, Jul 01 2023
%E A364003 a(23) corrected by _Charles R Greathouse IV_, Jul 03 2023
%E A364003 a(36)-a(45) from _Charles R Greathouse IV_, Jul 03 2023