1,1

Phi[P[n]]-P[Phi[n]]=A000010[A006530(n)]- A006530[A000010(n)]=-1, where P[x]=largest prime factor of x. Value of commutator of Phi and P functions at n equals -1.

Equivalently, n such that n and Phi(n) have the same largest prime factor since Phi(p)=p-1 if p is prime. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 08 2002

Same as the list of numbers n divisible by P(n)^2 (by Cloitre's comment, as P(n) = P(Phi(n)) if and only if P(n)^2 divides n). This divisibility implies that n cannot divide P(n)!, so A057109 is a supersequence. Hence all A002034(a(n)) are composite. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 28 2004

I. Kastanas, The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."

Eric Weisstein's World of Mathematics, GreatestPrimeFactor

Eric Weisstein's World of Mathematics, TotientFunction

Prime powers are in the sequence: n=p^c, P[n]=p, A=Phi[P[n]]=p-1, Phi[p^c]=[p^(c-1)]*(p-1), B=P[Phi[n]]=p, thus B-A=-1; Also n=507=3.13.13 is in the sequence, where the exponent of largest prime factor >1, so it remains maximal also prime factor of totient: Phi[P[507]]-P[Phi[507]=12-13=-1.

for(n=3, 1000, if(component(component(factor(n), 1), omega(n))==component(component(factor(eulerphi(n)), 1), omega(eulerphi(n))), print1(n, ", ")))

Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, -1], Print[n]], {n, 3, 10000}]

Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070004, A007283, A070813-A070816.

See also A057109, A002034, A102067, A102068.

Sequence in context: A086368 A034024 A140269 this_sequence A073539 A090779 A166402

Adjacent sequences: A070000 A070001 A070002 this_sequence A070004 A070005 A070006

nonn

Labos E. (labos(AT)ana.sote.hu), May 07 2002