A364003 Integers K such that PSL_2(K) is a K_4-simple group, i.e., |PSL_2(K)| has 4 distinct prime divisors.
11, 13, 16, 19, 23, 25, 27, 31, 32, 37, 47, 49, 53, 73, 81, 97, 107, 127, 128, 163, 193, 243, 257, 383, 487, 577, 863, 1153, 2187, 2593, 2917, 4373, 8192, 8747, 131072, 524288, 995327, 1492993, 1594323, 1990657, 5308417, 28311553, 86093443, 2147483648, 6879707137
Offset: 1
Keywords
Examples
|PSL_2(11)| = 660 = 2^2 * 3 * 5 * 11.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..81
- Y. Bugeaud, Z. Cao, and M. Mignotte, On Simple K4-Groups, Journal of Algebra, 241 (2001), 658-668.
Programs
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GAP
is:=function(n) return IsPrimePowerInt(n) and Length(Unique(FactorsInt(n*(n^2-1))))=4; end; Filtered([2..1000], n -> is(n)); # Charles R Greathouse IV, Jul 03 2023; edited by Lixin Zheng, Jun 23 2024
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PARI
is(n)=isprimepower(n) && omega(lcm([n-1,n,n+1]))==4 \\ Charles R Greathouse IV, Jul 03 2023
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PARI
H(n)=isprimepower(n/2^valuation(n,2)/3^valuation(n,3)) 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
Extensions
a(23) corrected by Charles R Greathouse IV, Jul 03 2023
a(36)-a(45) from Charles R Greathouse IV, Jul 03 2023
Comments