A381498 a(n) = sum of numbers k <= n that have the same squarefree kernel as n.

1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 36, 19, 30, 21, 22, 23, 60, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 96, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 144, 56, 120, 51, 78, 53, 198, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66

Offset: 1

Author

Michael De Vlieger, Mar 03 2025

Keywords

• nonn

Comments

Analogous to A244974(n) = sum of row n of A162306; row n of A369609 is a proper subset of A162306.

Examples

			 n  a(n)  Factor(a(n))  Row n of A369609
----------------------------------------
 4    6   2 * 3         {2, 4}
 8   14   2 * 7         {2, 4, 8}
 9   12   2^2 * 3       {3, 9}
12   18   2 * 3^2       {6, 12}
16   30   2 * 3 * 5     {2, 4, 8, 16}
18   36   2^2 * 3^2     {6, 12, 18}
20   30   2 * 3 * 5     {10, 20}
24   60   2^2 * 3 * 5   {6, 12, 18, 24}
25   30   2 * 3 * 5     {5, 25}
27   39   3 * 13        {3, 9, 27}
28   42   2 * 3 * 7     {14, 28}
32   62   2 * 31        {2, 4, 8, 16, 32}
36   96   2^5 * 3       {6, 12, 18, 24, 36}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• 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 other numbers in blue.

Crossrefs

Cf. A007947, A008479, A244974, A369609.

Programs

Mathematica

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

PARI

rad(n) = factorback(factorint(n)[, 1]);
a(n) = my(r=rad(n)); sum(k=1, n, if(rad(k)==r, k)); \\ Michel Marcus, Mar 03 2025

Formula

a(n) = sum of row n of A369609.

For squarefree k, a(k) = k.

For prime power p^m, a(p^m) = Sum_{i=1..m} p^i.