A391625 Squarefree kernel of the sum of numbers k <= n such that the squarefree kernel of k equals the squarefree kernel of n.

1, 2, 3, 6, 5, 6, 7, 14, 6, 10, 11, 6, 13, 14, 15, 30, 17, 6, 19, 30, 21, 22, 23, 30, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 6, 37, 38, 39, 70, 41, 42, 43, 66, 30, 46, 47, 6, 14, 30, 51, 78, 53, 66, 55, 14, 57, 58, 59, 30, 61, 62, 42, 42, 65, 66, 67, 102

Offset

1,2

Comments

Squarefree kernel of the sum of row n of A369609.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing a(n) for prime n in red, squarefree composite n in green, proper prime powers n in gold, powerful n that are not prime powers in magenta, and numbers neither squarefree nor powerful in blue.

Formula

a(n) = A007947(A381498(n)).

a(n) = n for squarefree n.

a(p^m) != p for prime p and m > 1, since A381498(p^m) = Sum_{i=1..m} p^i = p^A000217(m).

a(2^m) = 2^m - 2 for m > 1.

Example

Let s = A381498.
Table of n, a(n) for select n:
 n  a(n)  rad(s(n))  Row n of A369609
-------------------------------------
 1    1 = rad(1)   = {1}
 2    2 = rad(2)   = {2}
 3    3 = rad(3)   = {3}
 4    6 = rad(6)   = {2, 4}
 5    5 = rad(5)   = {5}
 6    6 = rad(6)   = {6}
 8   14 = rad(14)  = {2, 4, 8}
 9    6 = rad(12)  = {3, 9}
12    6 = rad(18)  = {6, 12}
18    6 = rad(36)  = {6, 12, 18}
20   30 = rad(30)  = {10, 20}
75   15 = rad(135) = {15, 45, 75}

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; rad[Total[Select[Range[n], rad[#] == r &]]], {n, 120}]

Crossrefs

Cf. A005117, A007947, A008479, A369609, A381498.

Sequence in context: A369319 A336465 A340774 * A345068 A057723 A381498

Adjacent sequences: A391622 A391623 A391624 * A391626 A391627 A391628

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 10 2026

Status

approved