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

Author

N. J. A. Sloane, Feb 26 2017

Keywords

• nonn

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.

Crossrefs

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

Programs

Magma

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

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 *)

Extensions

More terms from Vincenzo Librandi, Feb 28 2017