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%I #28 Dec 28 2020 17:46:27
%S 1,4,8,16,32,36,72,144,216,288,432,864,1728,2592,3600,5400,7200,10800,
%T 21600,43200,64800,108000,129600,216000,259200,324000,529200,1058400,
%U 2116800,3175200,5292000,6350400,10584000,12700800,15876000,31752000,63504000,95256000
%N 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).
%C This sequence is infinite.
%C 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.
%C 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
%H Amiram Eldar, <a href="/A283052/b283052.txt">Table of n, a(n) for n = 1..46</a>
%e 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.
%t uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@
%t 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
%o (PARI) uphi(n) = my(f = factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1);
%o 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
%Y Cf. A000010, A047994, A057723, A285906.
%Y Similar sequences: A002110, A004394, A037992, A061742, A292984, A329882, A333953.
%K nonn
%O 1,2
%A _Amiram Eldar_, May 19 2017
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