%I #19 Sep 17 2024 04:03:55

%S 1,5,9,11,13,17,19,23,25,27,29,32,37,41,43,47,49,53,59,61,64,65,67,69,

%T 71,73,77,79,81,83,89,97,101,103,104,107,109,113,119,121,125,128,129,

%U 131,137,139,149,151,153,155,157,163,164,167,169,173,179,181,185

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

%C This sequence, alongside A006049 and A294278, form a partition of the positive integers.

%C The asymptotic density of this sequence is 1/2 (Erdős, 1936). - _Amiram Eldar_, Sep 17 2024

%H Amiram Eldar, <a href="/A294277/b294277.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 omega(1) = 0 < omega(2) = 1, hence 1 belongs to this sequence.

%e omega(4) = 1 = omega(5) = 1, hence 4 does not belong to this sequence.

%e omega(6) = 2 > omega(7) = 1, hence 6 does not belong to this sequence.

%t Position[Partition[PrimeNu[Range[200]],2,1],_?(#[[1]]<#[[2]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale_, May 06 2018 *)

%o (PARI) for (n=1, 185, if (omega(n) < omega(n+1), print1 (n ", ")))

%Y Cf. A001221, A006049, A294278.

%K nonn,easy

%O 1,2

%A _Rémy Sigrist_, Oct 26 2017