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A343783
a(n) is the largest primorial number (A002110) which divides phi(n).
1
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 29 2021
STATUS
approved