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

Author

Amiram Eldar, Jan 12 2024

Keywords

• nonn
• easy

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

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

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);}