A369020 - OEIS

%I #11 Sep 11 2024 13:39:42

%S 2,5,6,10,13,14,21,22,29,30,33,34,37,38,41,42,44,46,49,57,58,61,65,66,

%T 69,70,73,75,77,78,80,82,85,86,93,94,98,99,101,102,105,106,109,110,

%U 113,114,116,118,122,129,130,133,135,137,138,141,142,145,147,154,157

%N Numbers k such that k and k+1 have the same maximal exponent in their prime factorization.

%C Differs from A358817 by having the terms 99, 165, 166, ..., which are not in A358817, and not having the terms 1, 440, 1331, 1575, ..., which are in A358817.

%C Numbers k such that A051903(k) = A051903(k+1).

%C If k is a term then k*(k+1) is a term of A362605.

%C The asymptotic density of this sequence is d(2) + Sum_{k>=2} (d(k) + d(k+1) - 2 * d2(k)) = 0.36939178586283962461..., where d(k) = Product_{p prime} (1 - 2/p^k) and d2(k) = Product_{p prime} (1 - 1/p^k - 1/p^(k+1)).

%H Amiram Eldar, <a href="/A369020/b369020.txt">Table of n, a(n) for n = 1..10000</a>

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

%o (PARI) emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));

%o lista(kmax) = {my(e1 = 0, e2); for(k = 2, kmax, e2 = emax(k); if(e1 == e2, print1(k-1, ", ")); e1 = e2);}

%Y Cf. A051903, A358817, A362605, A369022.

%Y Subsequences: A007674, A071318, A369021.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Jan 12 2024