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A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

%I A376847 #17 Oct 26 2024 04:22:54
%S A376847 0,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,1,0,3,1,1,0,4,0,1,0,3,0,4,0,0,1,1,
%T A376847 1,2,0,1,1,5,0,5,0,3,2,1,0,6,0,1,1,3,0,1,1,5,1,1,0,11,0,1,2,0,1,5,0,3,
%U A376847 1,5,0,4,0,1,1,3,1,5,0,8,0,1,0,11,1,1,1
%N A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.
%H A376847 Michael De Vlieger, <a href="/A376847/b376847.txt">Table of n, a(n) for n = 1..10000</a>
%H A376847 Michael De Vlieger, <a href="/A376847/a376847.png">Hasse diagrams of m in select R(n)</a>, where R(n) is the union of rows n of A162306 and A376248, indicating in blue those m > n such that Omega(m) <= Omega(n).
%F A376847 a(n) = card({m > n : rad(m) | n, Omega(m) <= Omega(n) }).
%F A376847 a(n) = 0 for prime power n (in A000961).
%F A376847 a(n) = card(A376248 \ A162306).
%F A376847 a(n) = A376567(n) - A010846(n) + A376546(n) = binomial(A001222(n) + A001221(n), A001221(n)) - A010846(n) + A376546(n).
%e A376847 Table of select n such that a(n) > 0:
%e A376847    n  a(n)  List of m in A376248 such that Omega(m) <= Omega(n)
%e A376847   -------------------------------------------------------------
%e A376847    6    1   {9}
%e A376847   10    1   {25}
%e A376847   12    2   {18, 27}
%e A376847   14    1   {49}
%e A376847   15    1   {25}
%e A376847   18    1   {27}
%e A376847   20    3   {25, 50, 125}
%e A376847   24    4   {27, 36, 54, 81}
%e A376847   28    3   {49, 98, 343}
%e A376847   30    4   {45, 50, 75, 125}
%e A376847   40    5   {50, 100, 125, 250, 625}
%e A376847   48    6   {54, 72, 81, 108, 162, 243}
%e A376847   60   11   {75, 81, 90, 100, 125, 135, 150, 225, 250, 375, 625}
%p A376847 with(NumberTheory):
%p A376847 cond := (m, n) -> irem(n, Radical(m)) = 0 and Omega(m) <= Omega(n):
%p A376847 a := n -> nops(select(m -> cond(m, n), [seq(n+1..A068795(n))])):
%p A376847 seq(a(n), n = 1..87);  # _Peter Luschny_, Oct 25 2024
%t A376847 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A376847 Table[k = PrimeOmega[n]; w = PrimeNu[n]; Binomial[k + w, w] - Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)], {n, 120}]
%Y A376847 Cf. A000961, A001221, A001222, A007947, A162306, A376567, A376846, A068795.
%K A376847 nonn
%O A376847 1,12
%A A376847 _Michael De Vlieger_, Oct 13 2024