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%I #51 Jul 08 2023 20:14:41
%S 8,13,14,18,19,20,25,36,38,43,48,51,52,54,60,71,74,75,78,80,87,91,92,
%T 105,108,110,112,114
%N Numbers k such that m(k!) = 2, where m(k) = mu(k) + mu(k+1) + mu(k+2).
%C Equivalently, k such that m(k!) = 2, where m(k) = mu(k+1) + mu(k+2), as mu(k!)=0 for all k >= 4 (because 4=2^2 divides k!). - _Rick L. Shepherd_, Aug 20 2003
%C 127 belongs to the sequence. - _Serge Batalov_, Feb 17 2011
%H Dario A. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.
%H Paul Leyland, <a href="https://web.archive.org/web/20191208132430/http://www.leyland.vispa.com:80/numth/factorization/factors/factorial+.txt">Factors of n!+1</a> (updated 1 Oct 2006).
%e 8 is a term: 8! = 40320; mu(40320) = 0, mu(40321) = 1, mu(40322) = 1, 0+1+1 = 2.
%e 98 is not a term because 98! + 2 = 2 * 31003012014959 * 114951592532951 * 2015644865638913835753087050212028452990938458387 * P78 has an odd number of factors. - _Sean A. Irvine_, Feb 03 2010
%o (PARI) m(n) = moebius(n)+moebius(n+1)+moebius(n+2); for(n=1,10^4, if(m(n!)==2,print(n)))
%Y Cf. A063838, A008683.
%Y Cf. A084846 (mu(n!+1)).
%K more,nonn
%O 1,1
%A _Jason Earls_, Aug 22 2001
%E More terms from _Rick L. Shepherd_, Aug 20 2003
%E Two more terms from _Sean A. Irvine_, Feb 03 2010, Feb 08 2010
%E Two new terms, 105 and 108, from _Daniel M. Jensen_, Feb 19 2011, Mar 02 2011
%E Two more terms, 110 and 112, from _Serge Batalov_, Mar 04-05 2011
%E One more term, 114, from _Sean A. Irvine_, May 25 2015
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