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.

45, 7, 11, 143, 13, 23, 119, 19, 667, 713, 29, 47, 31, 6929, 59, 407, 37, 41, 2867, 53, 83, 3149, 164561, 3233, 1403, 25631, 107, 61, 3763, 1633, 1679, 71, 79, 803, 73, 5959, 4559, 4717, 89, 4841, 36461, 167, 103, 5353, 179, 1067, 97, 101, 2507, 5989

Offset: 1

Author

Labos Elemer, May 10 2002

Keywords

• nonn
• look

Comments

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000215, A006530, A070812, A070813.

Programs

Maple

V:= Vector(100): count:= 0:
gpf:= t -> max(numtheory:-factorset(t)):
for m from 3 while count < 100 do
  v:= numtheory:-phi(gpf(m))-gpf(numtheory:-phi(m));
  if v::even or v < 1 or  v > 199 or  V[(v+1)/2] > 0 then next fi;
  V[(v+1)/2]:= m; count:= count+1;
od:
convert(V,list); # Robert Israel, Jun 24 2025

Mathematica

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

Formula

a(n) = min{x: phi(gpf(x)) - gpf(phi(x)) = 2n - 1} = min{x: A000010(A006530(x)) - A006530(A000010(x)) = 2n - 1}.

Extensions

5 and 17 removed to make name accurate by Sean A. Irvine, Jun 13 2024