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A376846 Number of m <= n such that rad(m) | n and Omega(m) > Omega(n), where rad = A007947 and Omega = A001222.

%I A376846 #12 Nov 07 2024 02:09:40
%S A376846 0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,2,0,0,0,2,0,1,0,2,0,0,1,3,
%T A376846 0,1,0,3,1,1,0,4,0,2,0,3,0,0,0,3,1,2,0,2,0,1,1,3,0,2,0,3,0,0,0,7,0,3,
%U A376846 1,5,0,1,0,4,0,3,0,8,0,1,0,4,0,4,0,4,2
%N A376846 Number of m <= n such that rad(m) | n and Omega(m) > Omega(n), where rad = A007947 and Omega = A001222.
%C A376846 Number of m not exceeding n such that the squarefree kernel of m divides n, and m has more prime factors with repetition than does n.
%C A376846 Number of m in row n of A162306 such that Omega(m) > Omega(n).
%H A376846 Michael De Vlieger, <a href="/A376846/b376846.txt">Table of n, a(n) for n = 1..10000</a>
%H A376846 Michael De Vlieger, <a href="/A376846/a376846.png">Hasse diagrams</a> of m in select rows n of A162306 indicating in red those m such that Omega(m) > Omega(n).
%H A376846 Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.18645.64480">Numbers k for which floor(log k / log lpf(k)) <= bigomega(k)</a>, 2024, about zeros in this sequence.
%F A376846 a(n) = card({m <= n : rad(m) | n, Omega(m) > Omega(n) }).
%F A376846 a(n) = 0 for prime power n (in A000961).
%F A376846 a(n) < A010846(n).
%e A376846 Table of select n such that a(n) > 0:
%e A376846    n  a(n)  List of m such that Omega(m) > Omega(n).
%e A376846   -------------------------------------------------
%e A376846   10   1    {8}
%e A376846   14   1    {8}
%e A376846   18   1    {16}
%e A376846   20   1    {16}
%e A376846   22   2    {8, 16}
%e A376846   26   2    {8, 16}
%e A376846   28   1    {16}
%e A376846   30   2    {16, 24}
%e A376846   33   1    {27}
%e A376846   34   3    {8, 16, 32}
%e A376846   36   1    {32}
%e A376846   38   3    {8, 16, 32}
%e A376846   39   1    {27}
%e A376846   40   1    {32}
%e A376846   42   4    {16, 24, 32, 36}
%t A376846 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A376846 {0}~Join~Table[With[{k = PrimeOmega[n]}, Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)]], {n, 2, 120}]
%Y A376846 Cf. A000961, A001222, A007947, A010846, A162306.
%K A376846 nonn
%O A376846 1,22
%A A376846 _Michael De Vlieger_, Oct 06 2024