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.

Showing 1-3 of 3 results.

A348198 Terms of A326835 having more divisors than any smaller term.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 225, 405, 495, 1155, 3675, 4455, 8085, 19635, 62475, 75735, 137445, 373065, 1187025, 1741905, 2611455, 8580495, 27301575, 50515245, 60063465, 248834355, 1021078905, 2374216515, 2822982855, 11695214685, 47990708535
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2021

Keywords

Comments

All the terms are odd since all the terms of A326835 are odd (as phi(1) = phi(2) = 1).
The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, ...

Examples

			The sequence A326835 begins with 1, 3, 5, 7, 9, 11, 13 and 15. The number of divisors of these terms are 1, 2, 2, 2, 3, 2, 2 and 4, respectively. The record values, 1, 2, 3 and 4, occur at 1, 3, 9 and 15, the first 4 terms of this sequence.

Crossrefs

Cf. A326835, A328858, A348199.

Programs

  • Mathematica
    q[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])] == Length[d]; dm = 0; s = {}; Do[If[q[n], d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[s, n]]], {n, 1, 10^6, 2}]; s

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

A348199 a(n) is the least term of A326835 whose number of divisors is n.

Original entry on oeis.org

1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 495, 531441, 3645, 2025, 1155, 43046721, 3675, 387420489, 4455, 18225, 295245, 31381059609, 8085, 50625, 2657205, 65025, 40095, 22876792454961, 34425, 205891132094649, 19635, 1476225, 215233605, 455625, 62475
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2021

Keywords

Comments

First differs from A038547 at n = 12.
All the terms are odd since all the terms of A326835 are odd (as phi(1) = phi(2) = 1).
a(n) exists for any n since 3^(n-1) is a term of A326835 which has n divisors.

Crossrefs

Cf. A038547, A326835, A328858, A348198.

Programs

  • Mathematica
    seq[m_] := Module[{p = Select[Range[m], PrimeQ], s = Table[0, {m}], c, nd, ndd}, s[[p]] = 3^(p - 1); c = Length[p]; n = 1; While[c < m, nd = DivisorSigma[0, n]; If[nd <= m && s[[nd]] == 0, ndd = Length@Union[EulerPhi /@ Divisors[n]]; If[ndd == nd, c++; s[[nd]] = n]]; n++]; s]; seq[30]

Formula

a(n) <= 3^(n-1), with equality if n is prime.
a(n) >= A038547(n).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.