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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.
0
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
1,1
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.
FORMULA
a(n) = min{x: phi(gpf(x)) - gpf(phi(x)) = 2n - 1 = min{x: A000010(A006530(x)) - A006530(A000010(x)) = 2n - 1}.
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, May 10 2002
EXTENSIONS
5 and 17 removed to make name accurate by Sean A. Irvine, Jun 13 2024
STATUS
approved