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A294278 Numbers k such that omega(k) > omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).

%I A294278 #15 Sep 17 2024 04:03:48
%S A294278 6,10,12,15,18,22,24,26,28,30,36,40,42,46,48,52,58,60,63,66,70,72,78,
%T A294278 80,82,84,88,90,96,100,102,105,106,108,110,112,114,120,124,126,130,
%U A294278 132,136,138,140,148,150,154,156,162,165,166,168,170,172,174,178,180
%N A294278 Numbers k such that omega(k) > omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).
%C A294278 The asymptotic density of this sequence is 1/2 (Erdős, 1936). - _Amiram Eldar_, Sep 17 2024
%H A294278 Amiram Eldar, <a href="/A294278/b294278.txt">Table of n, a(n) for n = 1..10000</a>
%H A294278 Paul Erdős, <a href="https://doi.org/10.1017/S0305004100019277">On a problem of Chowla and some related problems</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; <a href="https://www.renyi.hu/~p_erdos/1936-03.pdf">alternative link</a>.
%e A294278 omega(1) = 0 < omega(2) = 1, hence 1 does not belong to this sequence.
%e A294278 omega(4) = 1 = omega(5) = 1, hence 4 does not belong to this sequence.
%e A294278 omega(6) = 2 > omega(7) = 1, hence 6 belongs to this sequence.
%t A294278 Position[Differences[Array[PrimeNu, 200]], _?(# < 0 &)] // Flatten (* _Amiram Eldar_, Sep 17 2024 *)
%o A294278 (PARI) for (n=1, 180, if (omega(n) > omega(n+1), print1(n ", ")))
%Y A294278 Cf. A001221, A006049, A294277.
%K A294278 nonn,easy
%O A294278 1,1
%A A294278 _Rémy Sigrist_, Oct 26 2017