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.

A369021 Numbers k such that k, k+1 and k+2 have the same maximal exponent in their prime factorization.

Original entry on oeis.org

5, 13, 21, 29, 33, 37, 41, 57, 65, 69, 77, 85, 93, 98, 101, 105, 109, 113, 129, 137, 141, 157, 165, 177, 181, 185, 193, 201, 209, 213, 217, 221, 229, 237, 253, 257, 265, 281, 285, 301, 309, 317, 321, 329, 345, 353, 357, 365, 381, 389, 393, 397, 401, 409, 417, 429
Offset: 1

Views

Author

Amiram Eldar, Jan 12 2024

Keywords

Comments

Numbers k such that A051903(k) = A051903(k+1) = A051903(k+2).
The asymptotic density of this sequence is d(2,3) + Sum_{k>=2} (d(k+1,3) - d(k,3) + 3*d2(k,2,1) - 3*d2(k,1,2)) = 0.13122214221443994377..., where d(k,m) = Product_{p prime} (1 - m/p^k) and d2(k,m1,m2) = Product_{p prime} (1 - m1/p^k - m2/p^(k+1)).

Links

Crossrefs

Cf. A051903, A369022.
Subsequence of A369020.
Subsequences: A007675, A071319.

Programs

  • Mathematica
    emax[n_] := emax[n] = Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; Select[Range[200], emax[#] == emax[# + 1] == emax[#+2] &]
  • PARI
    emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));
lista(kmax) = {my(e1 = 0, e2 = 0, e3); for(k = 3, kmax, e3 = emax(k); if(e1 == e2 && e2 == e3, print1(k-2, ", ")); e1 = e2; e2 = e3);}

A369022 a(n) is the least start of a run of exactly n consecutive integers with the same maximal exponent in their prime factorization, or -1 if no such run exists.

Original entry on oeis.org

1, 2, 5, 844, 30923, 671346, 8870025
Offset: 1

Views

Author

Amiram Eldar, Jan 12 2024

Keywords

Comments

a(8) > 3.7*10^10.
a(8) <= 1770019255373287038727484868192109228824 which is the conjectured value of A219452(8)+1. - Giorgos Kalogeropoulos, Jan 15 2024

Crossrefs

Cf. A051903, A369020, A369021.
Similar sequences: A071125, A219452, A323253.

Programs

  • Mathematica
    emax[n_] := Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; ind = Position[Differences[Table[emax[n], {n, 1, 10^6}]], _?(# != 0 &)] // Flatten; d = Differences[ind]; seq = {1}; Do[i = FirstPosition[d, k]; If[MissingQ[i], Break[]]; AppendTo[seq, ind[[i[[1]]]] + 1], {k, 2, Max[d]}]; seq
  • PARI
    emax(n) = vecmax(factor(n)[, 2]);
lista(len) = {my(v = vector(len), w = [0], m, c = 0, k = 2); while(c < len, e = emax(k); m = #w; if(e == w[m], w = concat(w, e), if(m < = len && v[m] == 0, v[m] = k-m; c++); w = [e]); k++); v;}

Formula

A051903(a(n)) >= k for 2^k <= n < 2^(k+1)-1.

A376141 The maximum exponent in the prime factorization of the numbers k such that k and k+1 have the same maximum exponent in their prime factorization.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Sep 11 2024

Keywords

Links

Crossrefs

Cf. A051903, A369020.

Programs

  • Mathematica
    emax[n_] := Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; With[{t = Table[emax[n], {n, 1, 500}]}, t[[Position[Differences[t], 0] // Flatten]]]
  • PARI
    emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));
lista(kmax) = {my(e1 = 0, e2); for(k = 2, kmax, e2 = emax(k); if(e1 == e2, print1(e1, ", ")); e1 = e2);}

Formula

a(n) = A051903(A369020(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (d(2) + Sum_{k>=2} k * (d(k) + d(k+1) - 2 * d2(k)))/d0 = 1.14396758638154735362..., where d(k) = Product_{p prime} (1 - 2/p^k), d2(k) = Product_{p prime} (1 - 1/p^k - 1/p^(k+1)), and d0 = 0.36939178586283962461... is the asymptotic density of A369020.
Showing 1-3 of 3 results.

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