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A327877 Numbers having an odd number of non-unitary prime factors.

%I A327877 #17 Jun 05 2020 06:05:15
%S A327877 4,8,9,12,16,18,20,24,25,27,28,32,40,44,45,48,49,50,52,54,56,60,63,64,
%T A327877 68,75,76,80,81,84,88,90,92,96,98,99,104,112,116,117,120,121,124,125,
%U A327877 126,128,132,135,136,140,147,148,150,152,153,156,160,162,164,168
%N A327877 Numbers having an odd number of non-unitary prime factors.
%C A327877 Differs from A190641(n) from n = 310 (900, the least number with 3 non-unitary prime factors, is in this sequence but not in A190641).
%C A327877 The asymptotic density of the numbers in this sequence is 0.338682... = 1 - A065493.
%H A327877 Amiram Eldar, <a href="/A327877/b327877.txt">Table of n, a(n) for n = 1..10000</a>
%H A327877 Eckford Cohen, <a href="https://doi.org/10.1090/S0002-9947-1964-0166181-5">Some asymptotic formulas in the theory of numbers</a>, Transactions of the American Mathematical Society, Vol. 112, No. 2, (1964), pp. 214-227.
%H A327877 Willy Feller and Erhard Tornier, <a href="https://doi.org/10.1007/BF01448890">Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe</a>, Mathematische Annalen, Vol. 107, No. 1 (1933), pp. 188-232.
%H A327877 I. J. Schoenberg, <a href="https://doi.org/10.1090/S0002-9947-1936-1501849-X">On asymptotic distributions of arithmetical functions</a>, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330.
%t A327877 A056170[n_] := Count[FactorInteger[n], {_, k_ /; k > 1}]; Select[Range[200], OddQ[A056170[#]] &] (* after _Jean-François Alcover_ at A056170 *)
%o A327877 (PARI) \\ here b(n) is A056170(n)
%o A327877 b(n)={my(f=factor(n)[, 2]); sum(i=1, #f, f[i]>1)}
%o A327877 { select(k->b(k)%2, [1..200]) } \\ _Andrew Howroyd_, Sep 28 2019
%Y A327877 Cf. A056170, A065493, A190641, A333634 (complement).
%K A327877 nonn
%O A327877 1,1
%A A327877 _Amiram Eldar_, Sep 28 2019