cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A348002 Numbers with a record number of distinct values of the unitary totient function applied to their unitary divisors (A348001).

Original entry on oeis.org

1, 3, 12, 60, 420, 660, 4620, 8580, 9240, 60060, 78540, 106260, 157080, 1021020, 1381380, 1492260, 1806420, 2762760, 2984520, 23483460, 34321980, 38798760, 46966920, 68643960, 681020340, 892371480, 1848483780, 1990674840, 2127962760, 3226266120
Offset: 1

Views

Author

Amiram Eldar, Sep 23 2021

Keywords

Comments

The corresponding record values are 1, 2, 4, 8, 13, 16, 26, 28, 32, 40, 50, 52, 64, 72, 80, 84, 100, 104, 128, 144, 168, 176, 200, 256, 288, 352, 360, 416, 424, 512, ...
This sequence is infinite since A348001 is unbounded: A348001(n) >= omega(n).
a(31) > 2*10^10.

Examples

			The first 12 terms of A348001(k) are 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 and 4. The record values, 1, 2 and 4, are obtained at k = 1, 3 and 12. Therefore, this sequence begins with 1, 3, 12.

Crossrefs

The unitary version of A328858.
Cf. A047994, A348001, A348003.

Programs

  • Mathematica
    f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := Length @ Union[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

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

Views

Author

Amiram Eldar, Oct 04 2021

Keywords

Comments

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.

Crossrefs

The unitary version of A348159.
Cf. A348003, A348004, A348173.

Programs

  • Mathematica
    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
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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