A070818 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.

45, 5, 7, 11, 143, 13, 23, 119, 17, 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.

Links

Table of n, a(n) for n=1..52.

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

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

Sequence in context: A131994 A037940 A037218 * A285893 A129536 A108909

Adjacent sequences: A070815 A070816 A070817 * A070819 A070820 A070821

Keyword

nonn

Author

Labos Elemer, May 10 2002

Status

approved