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 1-5 of 5 results.

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

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.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Sep 23 2021

Keywords

Comments

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.

Crossrefs

The unitary version of A328859.
Cf. A034444, A047994, A348001, A348002, A348004.

Programs

  • Mathematica
    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

A348004 Numbers whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91
Offset: 1

Views

Author

Amiram Eldar, Sep 23 2021

Keywords

Comments

First differs from A042965 \ {0} at n=63, and from A122906 at n=53.
Since Sum_{d|k, gcd(d,k/d)=1} uphi(d) = k, these are numbers k such that the set {uphi(d) | d|k, gcd(d,k/d)=1} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345), since an odd prime power p^e has 2 unitary divisors, 1 and p^e, whose uphi values are 1 and p^e - 1. It also includes all the powers of 2, except for 2 (A151821).
If k is a term, then all the unitary divisors of k are also terms.
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 74, 741, 7386, 73798, 737570, 7374534, 73740561, 737389031, 7373830133, ... Apparently, this sequence has an asymptotic density 0.73738...

Examples

			4 is a term since it has 2 unitary divisors, 1 and 4, and uphi(1) = 1 != uphi(4) = 3.
12 is a term since the uphi values of its unitary divisors, {1, 3, 4, 12}, are distinct: {1, 2, 3, 6}.

Links

Crossrefs

The unitary version of A326835.
Cf. A034444, A042965, A047994, A061345, A151821, A122906, A348001.

Programs

  • Mathematica
    f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])] == Length[d]; Select[Range[100], q]
  • Python
    from math import prod
from sympy.ntheory.factor_ import udivisors, factorint
A348004_list = []
for n in range(1,10**3):
    pset = set()
    for d in udivisors(n,generator=True):
        u = prod(p**e-1 for p, e in factorint(d).items())
        if u in pset:
            break
        pset.add(u)
    else:
        A348004_list.append(n) # Chai Wah Wu, Sep 24 2021

Formula

Numbers k such that A348001(k) = A034444(k).

A348173 a(n) is the sum of the distinct values obtained when the unitary totient function is applied to the unitary divisors of n.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 24, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 56, 57, 29, 59, 60, 61, 31, 63, 64, 65, 33, 67, 68, 69, 35, 71, 72, 73, 37, 75, 76, 77, 39, 79, 80, 81, 41, 83, 78
Offset: 1

Views

Author

Amiram Eldar, Oct 04 2021

Keywords

Comments

First differs from A283971 at n = 84.

Examples

			The unitary divisors of 18 are {1, 2, 9, 18} and their uphi values are {1, 1, 8, 8}. The set of distinct values is {1, 8} whose sum is 9. Therefore, a(18) = 9.

Links

Crossrefs

The unitary version of A348158.
Cf. A047994, A077610, A283971, A348001, A348004.

Programs

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

Formula

a(n) <= n, with equality if and only if n is in A348004.

A361923 Number of distinct values obtained when the infinitary totient function (A091732) is applied to the infinitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 2, 7, 4, 2, 4
Offset: 1

Views

Author

Amiram Eldar, Mar 30 2023

Keywords

Comments

First differs from A348001 at n = 27.

Links

Crossrefs

Cf. A077609, A091732.
Similar sequences: A319696, A348001.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
a[n_] := Length @ Union[iphi /@ idivs[n]]; Array[a, 100]
  • PARI
    iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
a(n) = {my(d = idivs(n)); #Set(apply(x->iphi(x), d));}
Showing 1-5 of 5 results.

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