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%I #30 Dec 30 2022 09:34:10
%S 1,1,2,1,3,2,4,1,3,4,5,2,6,5,6,1,7,3,8,4,7,8,9,2,9,10,5,6,10,7,11,1,
%T 11,12,13,3,12,14,15,4,13,8,14,9,10,16,15,2,17,11,18,12,16,5,19,6,20,
%U 21,17,7,18,22,13,1,23,14,19,15,24,16,20,3,21,25,17,18,26,19,22,4,8,27,23
%N Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.
%C Equivalently, the number of positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity. - _Gus Wiseman_, Dec 28 2018
%C There is a close relationship between a(n) and a(n^2). See A209934 for an exploratory quantification. - _Peter Munn_, Aug 04 2019
%H Alois P. Heinz, <a href="/A058933/b058933.txt">Table of n, a(n) for n = 1..20000</a>
%F Ordinal transform of A001222 (bigomega). - _Franklin T. Adams-Watters_, Aug 28 2006
%F If a(n) < a(3^A001222(2n)) = A078843(A001222(2n)) then a(2n) = a(n), otherwise a(2n) > a(n). - _Peter Munn_, Aug 05 2019
%e 3 is prime, so a(3)=2. 10 is 2-almost prime (semiprime), so a(10)=4.
%e From _Gus Wiseman_, Dec 28 2018: (Start)
%e Column n lists the a(n) positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity:
%e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
%e ---------------------------------------------------------------------
%e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
%e 2 3 4 5 6 9 7 8 11 10 14 13 12 17 18
%e 2 3 4 6 5 7 9 10 11 8 13 12
%e 2 4 3 5 6 9 7 11 8
%e 2 3 4 6 5 7
%e 2 4 3 5
%e 2 3
%e 2
%e (End)
%p p:= proc() 0 end:
%p a:= proc(n) option remember; local t;
%p t:= numtheory[bigomega](n);
%p p(t):= p(t)+1
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Oct 09 2015
%t p[_] = 0; a[n_] := a[n] = Module[{t}, t = PrimeOmega[n]; p[t] = p[t]+1]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 24 2017, after _Alois P. Heinz_ *)
%Y Positions of 1's are A000079.
%Y Cf. A001358, A014612, A014613, A014614.
%Y Cf. A000010, A000961, A001222, A006049, A045920, A061142, A067003, A078843, A209934, A302242, A322838, A322839, A322840.
%Y Equivalent sequence restricted to squarefree numbers: A340313.
%K easy,nonn
%O 1,3
%A _Naohiro Nomoto_, Jan 11 2001
%E Name edited by _Peter Munn_, Dec 30 2022
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