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

1,3

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.

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)

Example

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).

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]

Prog

(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

Crossrefs

Cf. A000010, A002110, A034386, A053589.

Sequence in context: A329814 A130754 A164126 * A261902 A163368 A151948

Adjacent sequences: A343780 A343781 A343782 * A343784 A343785 A343786

Keyword

nonn

Author

Amiram Eldar, Apr 29 2021

Status

approved