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%I #17 Jul 15 2018 13:06:55
%S 1,1,1,2,2,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N a(n) = the least k >= 1 such that n! + k is squarefree.
%C Conjecture: There exists a finite k such that a(n) < k for all n. Subsidiary sequence: Index of the first occurrence of n in this sequence. In case the conjecture is true, this sequence would be finite.
%C If a(n) = 2 ==> n!+1 is divisible by a square (sequence A064237). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004
%F a(n) = A092983(n) - n!.
%e a(5) = 2 = 122 - 5! = 122 - 120 (as 121 = 11^2 is not squarefree).
%t Table[SelectFirst[Range@ 10, SquareFreeQ[n! + #] &], {n, 45}] (* _Michael De Vlieger_, Aug 23 2017 *)
%o (PARI) a(n)=for(i=1,n!,if(issquarefree(n!+i),return(i)))
%o (PARI) A092984(n) = { my(k=1); while(!issquarefree(n!+k), k++); k; }; \\ _Antti Karttunen_, Aug 22 2017
%Y Cf. A000142, A064237, A092983.
%K nonn
%O 1,4
%A _Amarnath Murthy_, Mar 28 2004
%E More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004
%E More terms from _David Wasserman_, Sep 27 2006
%E Typo in description corrected by _Antti Karttunen_, Aug 22 2017
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