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
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