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

1,1

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

Adjacent sequences: A364000 A364001 A364002 * A364004 A364005 A364006

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