A372322 - OEIS

%I #6 May 11 2024 21:58:39

%S 1,2,5,6,5,11,18,8,16,22,5,28,13,33,23,38,11,26,12,9,58,28,80,5,30,55,

%T 19,27,19,56,37,21,27,87,44,44,48,38,18,58,42,5,110,26,112,140,38,45,

%U 32,144,102,59,5,139,225,39,44,22,180,86,114,34,23,133,41,115

%N a(n) = A010846(A372111(n)).

%C Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.

%C Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.

%C a(n) is the length of row A372111(n) of A162306.

%C Analogous to A371909, which instead regards A109890 and A109735.

%H Michael De Vlieger, <a href="/A372322/b372322.txt">Table of n, a(n) for n = 1..10000</a>

%e Let s(x) = A372111(x) and let r(x) = A010846(x).

%e a(1) = 1 since r(s(1)) = r(1) = 1.

%e a(2) = 2 since r(s(2)) = r(3) = 2. For prime p, r(p) = card({1, p}) = 2.

%e a(3) = 5 since r(s(3)) = r(6) = 5. r(6) = card({1, 2, 3, 4, 6}) = 5.

%e a(4) = 6 since r(s(4)) = r(10) = 6. r(10) = card({1, 2, 4, 5, 8, 10}) = 6.

%e a(5) = 5 since r(s(5)) = r(15) = 5. r(15) = card({1, 3, 5, 9, 15}) = 5.

%e a(6) = 11 since r(s(6)) = r(24) = 11. r(24) = card({1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24}) = 11, etc.

%t nn = 68; c[_] := False;

%t rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];

%t f[x_] := Select[Range[x], Divisible[x, rad[#]] &];

%t Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];

%t {1}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];

%t   Sow[Length[r]]; c[k] = True;

%t   s += k, {i, 3, nn}] ][[-1, 1]]

%Y Cf. A007947, A010846, A124652, A162306, A371909, A372111, A372323.

%K nonn

%O 1,2

%A _Michael De Vlieger_, May 05 2024