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A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994). 2
1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 864, 1728, 2592, 3600, 5400, 7200, 10800, 21600, 43200, 64800, 108000, 129600, 216000, 259200, 324000, 529200, 1058400, 2116800, 3175200, 5292000, 6350400, 10584000, 12700800, 15876000, 31752000, 63504000, 95256000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is infinite.

a(1) = 1, a(6) = 36, a(15) = 3600 and a(32) = 6350400 are the smallest numbers n such that uphi(n)/phi(n) = 1, 2, 3 and 4. They are squares of 1, 6, 60, and 2520.

Also, coreful superabundant numbers: numbers k with a record value of the coreful abundancy index, A057723)(k)/k > A057723)(m)/m for all m < k. The two sequences are equivalent since A057723)(k)/k = A047994(k)/A000010(k) for all k. - Amiram Eldar, Dec 28 2020

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..46

EXAMPLE

uphi(k)/phi(k) = 1, 1, 1, 3/2 for k = 1, 2, 3, 4, thus a(1) = 1 and a(2) = 4 since a(4) > a(m) for m < 4.

MATHEMATICA

uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@

FactorInteger[n]))[[1]]]; a = {}; rmax = 0; For[k = 0, k < 10^9, k++; r = uphi[k]/EulerPhi[k]; If[r > rmax, rmax = r; a = AppendTo[a, k]]]; a

PROG

(PARI) uphi(n) = my(f = factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2]-1);

lista(nn) = {my(rmax = 0); for (n=1, nn, if ((newr=uphi(n)/eulerphi(n)) > rmax, print1(n, ", "); rmax = newr); ); } \\ Michel Marcus, May 20 2017

CROSSREFS

Cf. A000010, A047994, A057723, A285906.

Similar sequences: A002110, A004394, A037992, A061742, A292984, A329882, A333953.

Sequence in context: A354955 A304250 A322793 * A088259 A123857 A217313

Adjacent sequences: A283049 A283050 A283051 * A283053 A283054 A283055

KEYWORD

nonn

AUTHOR

Amiram Eldar, May 19 2017

STATUS

approved

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Last modified January 30 03:32 EST 2023. Contains 359939 sequences. (Running on oeis4.)