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).

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

Offset: 1

Author

Amiram Eldar, May 19 2017

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000010, A047994, A057723, A285906.

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

Programs

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

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