A343783 a(n) is the largest primorial number (A002110) which divides phi(n).

1, 1, 2, 2, 2, 2, 6, 2, 6, 2, 2, 2, 6, 6, 2, 2, 2, 6, 6, 2, 6, 2, 2, 2, 2, 6, 6, 6, 2, 2, 30, 2, 2, 2, 6, 6, 6, 6, 6, 2, 2, 6, 6, 2, 6, 2, 2, 2, 6, 2, 2, 6, 2, 6, 2, 6, 6, 2, 2, 2, 30, 30, 6, 2, 6, 2, 6, 2, 2, 6, 2, 6, 6, 6, 2, 6, 30, 6, 6, 2, 6, 2, 2, 6, 2, 6

Offset: 1

Author

Amiram Eldar, Apr 29 2021

Keywords

• nonn

Examples

			a(3) = 2 since phi(3) = 2 and 2 = A002110(1).
a(5) = 2 since phi(5) = 4 and 2 = A002110(1) is the largest primorial dividing 4.
a(7) = 6 since phi(7) = 6 and 6 = A002110(2).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Pollack and Carl Pomerance, Phi, primorials, and Poisson, Illinois J. Math., Vol. 64, No. 3 (2020), pp. 319-330; alternative link.

Crossrefs

Cf. A000010, A002110, A034386, A053589.

Programs

Mathematica

prim[n_] := Times @@ Prime[Range[n]]; gp[n_] := Module[{k = 1}, While[Divisible[n, prim[k]], k++]; prim[k - 1]]; a[n_] := gp[EulerPhi[n]]; Array[a, 100]

PARI

f(n) = my(s=1); forprime(p=2, , if(n%p, return(s), s *= p)); \\ A053589
a(n) = f(eulerphi(n)); \\ Michel Marcus, May 01 2021

Formula

a(n) = A053589(A000010(n)).

Let pr(n) be the largest prime divisor of a(n) (i.e., a(n) = pr(n)# = A034386(pr(n))). Then pr(n) ~ log(log(n))/log(log(log(n))) on a set of integers of asymptotic density 1 (Pollack and Pomerance, 2020).

From Bernard Schott, May 05 2021: (Start)

a(2n) = a(n) for n>=1.

a(n) = 1 iff n = 1 or n = 2.

a(n) = 2 iff 3 does not divide phi(n) (A088232)

a(n) >= 6 iff 3 divides phi(n) (A066498). (End)