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

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, 58, 31, 96, 32, 65, 10, 35, 76, 38, 131, 39, 83, 84, 11, 88, 42, 156, 43, 97, 45, 174, 46, 104, 106, 12, 111, 50, 283, 206, 51, 121, 53, 228, 54, 130, 133, 13, 138, 58

Offset: 1

Author

Michael De Vlieger, Oct 24 2023

Keywords

• nonn

Comments

Not a permutation of natural numbers: a(4) = a(7) = 5.

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).

The number m appears after its factors in S_rad(m). If k < sqrt(m) then k^2 also appears before m.

Scatterplot exhibits trajectories according to t = omega(m) = A001221(A025487(n)). The first term in each trajectory is A002110(t).

Examples

			a(1) = 1 since 1 is the only number k that does not exceed 1 such that rad(k) | 1.
a(2) = 2 since k in {1, 2} are such that rad(k) | 2.
a(3) = 3 since k in {1, 2, 4} are such that rad(k) | 4.
a(4) = 5 since k in {1, 2, 3, 4, 6} are such that rad(k) | 6, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1809 (all terms a(n) <= A002110(10))
• Michael De Vlieger, Log log scatterplot of a(n), 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.

Crossrefs

Cf. A001221, A002110, A007947, A010846, A025487, A363061.

Programs

Mathematica

rad[x_] := Times @@ FactorInteger[#][[All, 1]];
Map[Function[{n, r},
Count[Range[n], _?(Divisible[r, rad[#]] &)]] @@ {#, rad[#]} &,
  {1}~Join~Select[Range[Times @@ Prime@ Range[6]],
     # == Transpose@ {Prime@ Range[Length[#]], ReverseSort[#[[All, -1]] ]} &@
     FactorInteger[#] &] ]

Formula

a(n) = A010846(A025487(n)).