A348173 -id:A348173 - cp's OEIS frontend

A348264 a(n) is the number of iterations that n requires to reach a fixed point under the map x -> A348173(x).

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

a(n) first differs from A011765(n+2) at n = 84.
The fixed points are terms of A348004, so a(n) = 0 if and only if n is a term of A348004.
Conjecture: essentially partial sums of A219977 (verified for n <= 5000).

			a(1) = 0 since 1 is in A348004.
a(2) = 1 since there is one iteration of the map x -> A348173(x) starting from 2: 2 -> 1.
a(84) = 2 since there are 2 iterations of the map x -> A348173(x) starting from 84: 84 -> 78 -> 39.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A047994, A011765, A348004, A348173.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := Plus @@ DeleteDuplicates[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; a[n_] := -2 + Length@ FixedPointList[s, n]; Array[a, 100]

A348174 Indices k of records of low value in the ratios A348173(k)/k.

Original entry on oeis.org

1, 2, 546, 2730, 13650, 51870, 101010, 199290, 505050, 881790, 996450, 1919190, 32626230, 140100870, 654443790, 865554690

Offset: 1

Amiram Eldar, Oct 04 2021

The maximal possible value of the ratio A348173(k)/k is 1 which occurs at the terms of A348004.
The rounded values of the corresponding records are 1, 0.5, 0.478, 0.469, 0.466, 0.465, 0.4642, 0.4638, 0.4621, 0.4620, 0.460, 0.453, 0.450, 0.447, 0.446, 0.445, ...
a(17) > 1.4*10^9.

The unitary version of A348159.

Cf. A348003, A348004, A348173.

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

cp's OEIS Frontend

A348264 a(n) is the number of iterations that n requires to reach a fixed point under the map x -> A348173(x).

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