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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.
2
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
1,1
COMMENTS
This sequence is conjectured to be infinite, see Bugeaud, Cao, & Mignotte.
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.
EXAMPLE
|PSL_2(11)| = 660 = 2^2 * 3 * 5 * 11.
PROG
(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
(PARI) is(n)=isprimepower(n) && omega(lcm([n-1, n, n+1]))==4 \\ Charles R Greathouse IV, Jul 03 2023
(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
CROSSREFS
Subsequence of A000961.
Cf. A003586.
Sequence in context: A254412 A215778 A211021 * A214746 A341162 A205693
KEYWORD
nonn
AUTHOR
Lixin Zheng, Jul 01 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
STATUS
approved