A362041 a(0) = 1; for n > 0, a(n) is the largest k < A013929(n) such that rad(k) = rad(A013929(n)), where rad(n) = A007947(n).

1, 2, 4, 3, 6, 8, 12, 10, 18, 5, 9, 14, 16, 24, 20, 22, 15, 36, 7, 40, 26, 48, 28, 30, 21, 32, 34, 54, 45, 38, 50, 27, 42, 44, 60, 46, 72, 56, 33, 80, 52, 96, 98, 58, 39, 90, 11, 62, 25, 84, 64, 66, 75, 68, 70, 108, 63, 74, 120, 76, 51, 78, 100, 144, 82, 126, 13, 57, 86, 35, 88, 150, 92, 94, 147, 162

Offset: 0

Author

Michael De Vlieger, May 01 2023

Keywords

• nonn

Comments

Permutation of natural numbers.

Let m = A013929(n) and let R_m be the sequence of numbers k such that rad(k) = rad(m). a(n) gives the predecessor of m in R_m.

Examples

			A013929(1) = 4; the smallest k < 4 such that rad(k) = rad(4) = 2 is a(1) = 2.
A013929(2) = 8; the smallest k < 8 such that rad(k) = rad(8) = 2 is a(2) = 4.
A013929(3) = 9; the smallest k < 9 such that rad(k) = rad(9) = 3 is a(3) = 3.
A013929(4) = 12; the smallest k < 12 such that rad(k) = 6 is a(4) = 6.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^16
• Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Numbers k with omega(k) > 1 and all exponents exceeding 1 are highlighted in large light blue dots.

Crossrefs

Cf. A007947, A013929, A289280.

Programs

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; {1}~Join~Table[Function[r, SelectFirst[Range[m - 1, 1, -1], r == rad[#] &] ][rad[m]], {m, Select[Range[225], Not @* SquareFreeQ]}]

Formula

A013929(n) = p^e, a prime power, e > 0, implies a(n) = p^(e-1).

A013929(n) = p^2 implies a(n) = p.