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%I #64 Jun 25 2023 17:33:04
%S 4,5,7,12,23
%N Numbers k such that k! + 1 is divisible by a square.
%C 229 is another term because 613^2 divides 229!+1. See A115091 for primes whose square divides m!+1 for some m. An examination of the factorizations of m!+1 for m<=100 found no additional squares. - _T. D. Noe_, Mar 01 2006
%C 562 is also a term because 562!+1 is divisible by 563^2. - _Vladeta Jovovic_, Mar 30 2004
%C Comment from Francois BRUNAULT, Nov 23 2008: A web search reveals that for 1 <= k <= 228 there are 82 values of k for which k! + 1 has not been completely factored (the smallest is k=103), so showing that 229 and 562 are indeed the next two terms will be a huge task. I checked that k!+1 is not divisible by p^2 for k <= 1000 and prime p < 10^8.
%C It is very likely that 229 and 562 are the next two terms, but this has not yet been proved. - Nov 29 2008
%C Contains A007540(n)-1 for all n. That sequence is conjectured to be infinite. - _Robert Israel_, Jul 04 2016
%C This sequence includes A146968 (solutions of Brocard's problem). - _Salvador Cerdá_, Mar 08 2016
%C If k > 562 and k! + 1 is divisible by p^2 where p is prime, then either k > 10000 or p > 2038074743 (the hundred millionth prime). - _Jason Zimba_, Oct 21 2021
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha104.htm">Factorizations of m!+1</a>
%e 4 is in the sequence because 4! + 1 = 5^2.
%e 5 is in the sequence because 5! + 1 = 11^2.
%e 6 is not in the sequence because 6! + 1 = 721
%e 7 is in the sequence because 7! + 1 = 71^2.
%e 12 is in the sequence because 12! + 1 = 13^2 * 2834329.
%e 23 is a term because 23!+1 = 47^2*79*148139754736864591.
%e 229 and 562 are terms because
%e 229!+1 = 613^2 * 38669 * 1685231 * 3011917759 * (417-digit composite)
%e 562!+1 = 563^2 * 64467346976659839517037 * 112870688711507255213769871 * 63753966393108716329397432599379239 * (1214-digit prime). - _Thomas Richard_, Aug 31 2021
%p remove(t -> numtheory:-issqrfree(t!+1), [$1..50]); # _Robert Israel_, Jul 04 2016
%t Flatten[Position[MoebiusMu[Range[30]!+1], 0]]; (* _T. D. Noe_, Mar 01 2006, Nov 21 2008 *)
%o (PARI) lista(nn) = for(n=1, nn, if(!issquarefree(n!+1), print1(n, ", "))); \\ _Altug Alkan_, Mar 08 2016
%Y Cf. A007540 (Wilson primes), A115091, A146968, A038507, A085692.
%K nonn,more
%O 1,1
%A _Vladeta Jovovic_, Sep 22 2001
%E Example corrected by _T. D. Noe_, Nov 26 2008
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