cp's OEIS Frontend

A364225 a(n) = number of k <= m such that rad(k) | m, where m = A025487(n) and rad(n) = A007947(n).

%I A364225 #70 Nov 19 2023 10:33:30
%S A364225 1,2,3,5,4,8,5,11,18,6,14,15,26,7,18,20,36,8,23,44,25,68,26,49,9,29,
%T A364225 58,31,96,32,65,10,35,76,38,131,39,83,84,11,88,42,156,43,97,45,174,46,
%U A364225 104,106,12,111,50,283,206,51,121,53,228,54,130,133,13,138,58
%N A364225 a(n) = number of k <= m such that rad(k) | m, where m = A025487(n) and rad(n) = A007947(n).
%C A364225 Not a permutation of natural numbers: a(4) = a(7) = 5.
%C A364225 Let S_rad(m) be the sequence { k : rad(k) | rad(m) }. This sequence gives the number of k <= rad(m). Seen another way, this sequence gives the position of m in S_rad(m).
%C A364225 The number m appears after its factors in S_rad(m). If k < sqrt(m) then k^2 also appears before m.
%C A364225 Scatterplot exhibits trajectories according to t = omega(m) = A001221(A025487(n)). The first term in each trajectory is A002110(t).
%H A364225 Michael De Vlieger, <a href="/A364225/b364225.txt">Table of n, a(n) for n = 1..1809</a> (all terms a(n) <= A002110(10))
%H A364225 Michael De Vlieger, <a href="/A364225/a364225.png">Log log scatterplot of a(n)</a>, n = 1..1809, accentuating primorial A025487(n) in red, noting a(n) above in black, and labeling n in blue italic below for the first 24 terms.
%F A364225 a(n) = A010846(A025487(n)).
%e A364225 a(1) = 1 since 1 is the only number k that does not exceed 1 such that rad(k) | 1.
%e A364225 a(2) = 2 since k in {1, 2} are such that rad(k) | 2.
%e A364225 a(3) = 3 since k in {1, 2, 4} are such that rad(k) | 4.
%e A364225 a(4) = 5 since k in {1, 2, 3, 4, 6} are such that rad(k) | 6, etc.
%t A364225 rad[x_] := Times @@ FactorInteger[#][[All, 1]];
%t A364225 Map[Function[{n, r},
%t A364225 Count[Range[n], _?(Divisible[r, rad[#]] &)]] @@ {#, rad[#]} &,
%t A364225   {1}~Join~Select[Range[Times @@ Prime@ Range[6]],
%t A364225      # == Transpose@ {Prime@ Range[Length[#]], ReverseSort[#[[All, -1]] ]} &@
%t A364225      FactorInteger[#] &] ]
%Y A364225 Cf. A001221, A002110, A007947, A010846, A025487, A363061.
%K A364225 nonn
%O A364225 1,2
%A A364225 _Michael De Vlieger_, Oct 24 2023