|
%I #15 Jan 04 2017 10:26:35
%S 45,5,7,11,143,13,23,119,17,19,667,713,29,47,31,6929,59,407,37,41,
%T 2867,53,83,3149,164561,3233,1403,25631,107,61,3763,1633,1679,71,79,
%U 803,73,5959,4559,4717,89,4841,36461,167,103,5353,179,1067,97,101,2507,5989
%N Smallest argument m such that commutator[phi(n), gpf(n)] = 2n-1, where phi(n) = A000010(n) and gpf(n) = A006530(n), the largest prime factor of n.
%C Only five (no more) even commutator values appear at the arguments of known Fermat primes. These are listed in A070813. Still 0 and -1 emerge: A070812(3) = 0 and A070812(4) = -1.
%F a(n) = min{x: phi(gpf(x)) - gpf(phi(x)) = 2n - 1 = min{x: A000010(A006530(x)) - A006530(A000010(x)) = 2n - 1}.
%e f(m) = A070812(m) = A000010(A006530(m)) - A006530(A000010(m)); f(m) = 1 appears first at m = 45: phi(45) = 24, gpf(24) = 3, gpf(45) = 5, phi(5) = 4, so a(1) = phi(5) - gpf(24) = 4 - 3 = 1; also a(255) = 3321377 = 97*97*353: because its largest p factor gpf = 353, phi(353) = 352, phi(3321377) = 3277824 = 1024*3*11*97, with max prime factor = 97. Thus a(255) = 352 - 97 = 255.
%t pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] f[x_] := EulerPhi[pf[n]]-pf[EulerPhi[n]] t=Table[0, {257}]; Do[s=f[n]; If[s<258&&t[[s]]==0, t[[s]]=n], {n, 3, 4000000}]; t
%Y Cf. A000010, A000215, A006530, A070812, A070813.
%K nonn
%O 1,1
%A _Labos Elemer_, May 10 2002
|
