A328859 -id:A328859 - cp's OEIS frontend

A348003 Indices k of records of low value of the ratios A348001(k)/A034444(k) between the number of distinct values of the unitary totient function applied to the unitary divisors of k and the number of unitary divisors of k.

1, 2, 546, 2730, 13650, 101010, 199290, 996450, 1919190, 7373730, 28020174, 32626230, 125353410, 140100870, 700504350, 2381714790, 11908573950, 15270994830

Offset: 1

Amiram Eldar, Sep 23 2021

The maximal possible value of the ratio A348001(k)/A034444(k) is 1 which occurs at the terms of A348004.
The rounded values of the corresponding record values are 1, 0.5, 0.438, 0.406, 0.375, 0.359, 0.344, 0.312, 0.281, 0.266, 0.258, 0.250, 0.242, 0.199, 0.195, 0.170, 0.168, 0.145, ...
a(19) > 2*10^10, if it exists.

The unitary version of A328859.

Cf. A034444, A047994, A348001, A348002, A348004.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; r[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])]/Length[d]; rm = 2; seq = {}; Do[r1 = r[n]; If[r1 < rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A348159 Indices k of records of low value in the ratios A348158(k)/k.

Original entry on oeis.org

1, 2, 126, 1638, 2394, 8190, 139230, 155610, 2645370, 5757570, 97878690, 420302610, 1963331370, 7145144370

Offset: 1

Amiram Eldar, Oct 03 2021

The maximal possible value of the ratio A348158(k)/k is 1 which occurs at the terms of A326835.
The rounded values of the corresponding records are 1, 0.5, 0.452, 0.445, 0.437, 0.424, 0.420, 0.409, 0.404, 0.398, 0.3933, 0.3927, 0.3885, 0.3879, ...
a(15) <= 33376633290. - David A. Corneth, Oct 04 2021

Cf. A326835, A328859, A348158.

r[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]/n; rm = 2; s = {}; Do[If[(r1 = r[n]) < rm, rm = r1; AppendTo[s, n]], {n, 1, 2*10^5}]; s

f(n) = vecsum(Set(apply(eulerphi, divisors(n)))); \\ A348158
lista(nn) = {my(r=oo, x); for (i=1, nn, if ((x=f(i)/i) < r, print1(i, ", "); r = x););} \\ Michel Marcus, Oct 04 2021

cp's OEIS Frontend

A348003 Indices k of records of low value of the ratios A348001(k)/A034444(k) between the number of distinct values of the unitary totient function applied to the unitary divisors of k and the number of unitary divisors of k.

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