cp's OEIS Frontend

A070818 Smallest argument m such that commutator[phi(m), gpf(m)] = 2n-1, where phi(m) = A000010(m) and gpf(m) = A006530(m), the largest prime factor of m.

%I A070818 #29 Jun 24 2025 03:30:13
%S A070818 45,7,11,143,13,23,119,19,667,713,29,47,31,6929,59,407,37,41,2867,53,
%T A070818 83,3149,164561,3233,1403,25631,107,61,3763,1633,1679,71,79,803,73,
%U A070818 5959,4559,4717,89,4841,36461,167,103,5353,179,1067,97,101,2507,5989
%N A070818 Smallest argument m such that commutator[phi(m), gpf(m)] = 2n-1, where phi(m) = A000010(m) and gpf(m) = A006530(m), the largest prime factor of m.
%C A070818 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.
%H A070818 Robert Israel, <a href="/A070818/b070818.txt">Table of n, a(n) for n = 1..10000</a>
%F A070818 a(n) = min{x: phi(gpf(x)) - gpf(phi(x)) = 2n - 1} = min{x: A000010(A006530(x)) - A006530(A000010(x)) = 2n - 1}.
%e A070818 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.
%p A070818 V:= Vector(100): count:= 0:
%p A070818 gpf:= t -> max(numtheory:-factorset(t)):
%p A070818 for m from 3 while count < 100 do
%p A070818   v:= numtheory:-phi(gpf(m))-gpf(numtheory:-phi(m));
%p A070818   if v::even or v < 1 or  v > 199 or  V[(v+1)/2] > 0 then next fi;
%p A070818   V[(v+1)/2]:= m; count:= count+1;
%p A070818 od:
%p A070818 convert(V,list); # _Robert Israel_, Jun 24 2025
%t A070818 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 A070818 Cf. A000010, A000215, A006530, A070812, A070813.
%K A070818 nonn,look
%O A070818 1,1
%A A070818 _Labos Elemer_, May 10 2002
%E A070818 5 and 17 removed to make name accurate by _Sean A. Irvine_, Jun 13 2024