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

%I #7 Jan 16 2024 06:57:56

%S 1,2,5,6,10,13,14,21,22,29,30,33,34,37,38,41,42,44,46,49,57,58,61,65,

%T 66,69,70,73,75,77,78,80,82,85,86,93,94,98,101,102,105,106,109,110,

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

%N Numbers k such that A005361(k) = A005361(k+1).

%C First differs from A358817 at n = 165.

%C 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.

%C 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... .

%H Amiram Eldar, <a href="/A369211/b369211.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%Y Cf. A005361, A219452, A358817, A369166.

%Y Subsequences: A007674, A052213, A085651, A335328.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Jan 16 2024