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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 28 12:58 EST 2020. Contains 331321 sequences. (Running on oeis4.)