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

1,2

Comments

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

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.

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.

Example

 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}

Mathematica

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

Prog

(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

Crossrefs

Cf. A007947, A008479, A244974, A369609.

Sequence in context: A391625 A345068 A057723 * A335835 A361480 A142151

Adjacent sequences: A381495 A381496 A381497 * A381499 A381500 A381501

Keyword

nonn,nice

Author

Michael De Vlieger, Mar 03 2025

Status

approved