A219452 -id:A219452 - cp's OEIS frontend

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.

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

Offset: 1

Amiram Eldar, Jan 12 2024

a(8) > 3.7*10^10.

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

Cf. A051903, A369020, A369021.

Similar sequences: A071125, A219452, A323253.

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

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;}

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

A369211 Numbers k such that A005361(k) = A005361(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Jan 16 2024

First differs from A358817 at n = 165.
First differs from A369166 at n = 558. a(558) = 1520 is the least term that is not in A369166. A369166(144273) = 397952 is the least term of A369166 that is not a term of this sequence.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 368, 3638, 36337, 363163, 3631569, 36315800, 363156839, 3631559150, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36315... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A219452, A358817, A369166.

Subsequences: A007674, A052213, A085651, A335328.

s[n_] := s[n] = Times @@ FactorInteger[n][[;; , 2]]; Select[Range[300], s[#] == s[# + 1] &]

lista(kmax) = {my(c1 = 1, c2); for(k = 2, kmax, c2 = vecprod(factor(k)[, 2]); if(c1 == c2, print1(k-1, ", ")); c1 = c2);}

cp's OEIS Frontend

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.

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