A071319 -id:A071319 - cp's OEIS frontend

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

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

Amiram Eldar, Jan 12 2024

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)).

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

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

emax[n_] := emax[n] = Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; Select[Range[200], emax[#] == emax[# + 1] == emax[#+2] &]

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);}

A071320 Least of four consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that {k, k+1, k+2, k+3} are in A067259.

Original entry on oeis.org

844, 1681, 8523, 8954, 10050, 10924, 11322, 17404, 19940, 22020, 23762, 24450, 25772, 27547, 30923, 30924, 33172, 34347, 38724, 39050, 39347, 40050, 47673, 47724, 47825, 49147, 54585, 55449, 57474, 58473, 58849, 58867, 59924, 62865

Offset: 1

Labos Elemer, May 29 2002

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 0, 1, 4, 57, 555, 5492, 55078, 551443, 5512825, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000551... . - Amiram Eldar, Jan 18 2023

			k = 844 is a term since 844 = 2^2*211, k+1 = 845 = 5*13^2, k+2 = 846 = 2*3^2*47, and k+4 = 847 = 7*11^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michael De Vlieger)

Subsequence of A067259, A071318 and A071319.

Cf. A051903, A063528.

With[{s = Select[Range[10^5], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ SequencePosition[t, {1, 1, 1}][[All, 1]]]@ Differences@ s] (* Michael De Vlieger, Jul 30 2017 *)

A051903(k) = A051903(k+1) = A051903(k+2) = A051903(k+3) = 2 when k is a term.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI