A282706 Smallest prime factor of A020549(n) = (n!)^2 + 1.

2, 2, 5, 37, 577, 14401, 13, 101, 17, 131681894401, 13168189440001, 1593350922240001, 101, 38775788043632640001, 29, 1344169, 149, 9049, 37, 710341, 41, 61, 337, 509, 384956219213331276939737002152967117209600000001, 941

Offset

0,1

Comments

By construction, for n >= 2, a(n) == 1 (mod 4) and a(n) > n.

From Robert Israel, Mar 08 2017: (Start)

a(n) = A020549(n) for n in A046029.

a(n) <= 2*n+1 if n is in A104636.

The first member of A104636 for which a(n) < 2*n+1 is 48.

a(a(n)-n-1) = a(n). (End)

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.

Links

Hugo Pfoertner, Table of n, a(n) for n = 0..74

Apoloniusz Tyszka, On ZFC-formulae phi(x) for which we know a non-negative integer n such that max({x, element of N, phi(x)}) <= n if the set {x, element of N, phi(x)} is finite, 2019.

Apoloniusz Tyszka, Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).

Andrew Walker, Table of factors of (n!)^2+1.

Maple

f:= proc(n) local a;
  a:= min(map(proc(t) if t[1]::integer then t[1] fi end proc, ifactors((n!)^2+1, easy)[2]));
if a = infinity then
   a:= traperror(timelimit(60, min(map(t -> t[1], ifactors((n!)^2+1)[2]))));
fi;
  a
end proc:
map(f, [$0..36]); # Robert Israel, Mar 08 2017

Mathematica

Join[{2}, Array[FactorInteger[(#!)^2 + 1][[1, 1]]&, {25}]] (* Vincenzo Librandi, Feb 28 2017 *)

Prog

(Magma) [2] cat [Min(PrimeFactors(Factorial(n)^2 + 1)):n in[1..25]]; // Vincenzo Librandi, Feb 28 2017

Crossrefs

Cf. A020549, A046029, A083340, A083341, A104636, A282705.

Sequence in context: A076658 A218270 A218122 * A301346 A020549 A196128

Adjacent sequences: A282703 A282704 A282705 * A282707 A282708 A282709

Keyword

nonn

Author

N. J. A. Sloane, Feb 26 2017

Extensions

More terms from Vincenzo Librandi, Feb 28 2017

Status

approved