A092984 a(n) = the least k >= 1 such that n! + k is squarefree.

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, 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, 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

Offset: 1

Author

Amarnath Murthy, Mar 28 2004

Keywords

• nonn

Comments

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.

If a(n) = 2 ==> n!+1 is divisible by a square (sequence A064237). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004

Examples

			a(5) = 2 = 122 - 5! = 122 - 120 (as 121 = 11^2 is not squarefree).
		

Crossrefs

Cf. A000142, A064237, A092983.

Programs

Mathematica

Table[SelectFirst[Range@ 10, SquareFreeQ[n! + #] &], {n, 45}] (* Michael De Vlieger, Aug 23 2017 *)
Table[Module[{k=1,c=n!},While[!SquareFreeQ[c+k],k++];k],{n,110}] (* Harvey P. Dale, Jul 14 2025 *)

PARI

a(n)=for(i=1,n!,if(issquarefree(n!+i),return(i)))

PARI

A092984(n) = { my(k=1); while(!issquarefree(n!+k), k++); k; }; \\ Antti Karttunen, Aug 22 2017

Formula

a(n) = A092983(n) - n!.

Extensions

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004

More terms from David Wasserman, Sep 27 2006

Typo in description corrected by Antti Karttunen, Aug 22 2017