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A369166 Numbers k such that A000688(k) = A000688(k+1).

%I A369166 #9 Jan 15 2024 09:49:48
%S A369166 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 A369166 66,69,70,73,75,77,78,80,82,85,86,93,94,98,101,102,105,106,109,110,
%U A369166 113,114,116,118,122,129,130,133,135,137,138,141,142,145,147,154,157
%N A369166 Numbers k such that A000688(k) = A000688(k+1).
%C A369166 First differs from A358817 at n = 165.
%C A369166 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 368, 3632, 36266, 362468, 3624664, 36246863, 362468411, 3624675258, ... . From these values the asymptotic density of this sequence, whose existence was proven by Erdős and Ivić (1987) (the constant c in the Formula section), can be empirically evaluated by 0.36246... .
%D A369166 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, pp. 475-476.
%H A369166 Amiram Eldar, <a href="/A369166/b369166.txt">Table of n, a(n) for n = 1..10000</a>
%H A369166 Paul Erdős and Aleksandar Ivić, <a href="http://combinatorica.hu/~p_erdos/1987-32b.pdf">The distribution of values of a certain class of arithmetic functions at consecutive integers</a>, Colloq. Math. Soc. János Bolyai, 51, Number Theory, Budapest, 1987, pp. 45-91. See p. 60.
%F A369166 The number of terms not exceeding x, N(x) = c * x + O(x^(3/4) * log(x)^4), where c > 0 is a constant (Erdős and Ivić, 1987).
%t A369166 Select[Range[300], FiniteAbelianGroupCount[#] == FiniteAbelianGroupCount[#+1] &]
%o A369166 (PARI) lista(kmax) = {my(c1 = 1, c2); for(k = 2, kmax, c2 = vecprod(apply(numbpart, factor(k)[, 2])); if(c1 == c2, print1(k-1, ", ")); c1 = c2);}
%Y A369166 Cf. A000688, A358817.
%Y A369166 Subsequences: A007674, A052213, A085651, A335328.
%K A369166 nonn,easy
%O A369166 1,2
%A A369166 _Amiram Eldar_, Jan 15 2024