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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A362487 Infinitary highly totient numbers: numbers k that have more solutions x to the equation iphi(x) = k than any smaller k, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

1, 6, 12, 24, 48, 96, 144, 240, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 120960, 172800, 207360, 241920, 345600, 362880, 414720, 483840, 725760, 967680, 1209600, 1451520, 1935360, 2419200, 2903040, 3628800
Offset: 1

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Author

Amiram Eldar, Apr 22 2023

Keywords

Comments

Indices of records of A362485.
The corresponding numbers of solutions are 2, 4, 6, 10, 14, 18, 22, ... (A362488).

Links

Crossrefs

Cf. A091732, A362484, A362485, A362486, A362488.
Similar sequences: A097942, A100827, A361968, A362183.

Programs

  • Mathematica
    solnum[n_] := Length[invIPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^4] (* using the function invIPhi from A362484 *)

A216152 A205957(n) where n is a nonprime number.

Original entry on oeis.org

1, 2, 12, 48, 144, 1440, 34560, 483840, 7257600, 58060800, 3135283200, 125411328000, 2633637888000, 57940033536000, 5562243219456000, 27811216097280000, 723091618529280000, 6507824566763520000, 364438175738757120000, 327994358164881408000000
Offset: 1

Views

Author

Peter Luschny, Sep 02 2012

Keywords

Comments

The distinct values of A205957. Partial products of A216153.
a(1),...,a(10) are highly totient numbers (A097942) and products of distinct factorials (A058295). The author conjectures that this is true in general.

Links

Crossrefs

Cf. A051451.

Programs

  • Mathematica
    A205957[n_] := Exp[-Sum[MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]];
Table[A205957[n], {n, 0, 30}] // DeleteDuplicates (* Jean-François Alcover, Jul 08 2019 *)
  • Sage
    # sorted(list(set([A205957(n) for n in (0..31)])))
def A216152_list(n) :
    C = filter(lambda k: not is_prime(k), (1..n))
    return [A205957(c) for c in C]
A216152_list(31)

Formula

a(n) = A205957(A018252(n)).

A362402 Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

1, 4, 48, 72, 216, 288, 864, 1440, 1728, 2880, 3456, 4320, 5184, 5760, 8640, 12096, 17280, 25920, 34560, 48384, 51840, 69120, 103680, 120960, 155520, 181440, 207360, 241920, 311040, 362880, 483840, 622080, 725760, 967680, 1088640, 1209600, 1451520, 2177280, 2903040
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

The value 0 appears in the range of A162296 for all squarefree numbers (A005117) and therefore it is excluded from this sequence.
The corresponding record values are in A362403.
Except for 1, a subsequence of A362401.

Links

Crossrefs

Cf. A162296, A362401, A362403.
Similar sequences: A097942, A100827, A145899, A238895.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {1}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[1]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(1, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }
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