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

%I A381498 #11 Jun 04 2025 10:38:48
%S A381498 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,
%T A381498 26,39,42,29,30,31,62,33,34,35,96,37,38,39,70,41,42,43,66,60,46,47,
%U A381498 144,56,120,51,78,53,198,55,98,57,58,59,90,61,62,84,126,65,66
%N A381498 a(n) = sum of numbers k <= n that have the same squarefree kernel as n.
%C A381498 Analogous to A244974(n) = sum of row n of A162306; row n of A369609 is a proper subset of A162306.
%H A381498 Michael De Vlieger, <a href="/A381498/b381498.txt">Table of n, a(n) for n = 1..16384</a>
%H A381498 Michael De Vlieger, <a href="/A381498/a381498.png">Log log scatterplot of a(n)</a>, 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.
%F A381498 a(n) = sum of row n of A369609.
%F A381498 For squarefree k, a(k) = k.
%F A381498 For prime power p^m, a(p^m) = Sum_{i=1..m} p^i.
%e A381498  n  a(n)  Factor(a(n))  Row n of A369609
%e A381498 ----------------------------------------
%e A381498  4    6   2 * 3         {2, 4}
%e A381498  8   14   2 * 7         {2, 4, 8}
%e A381498  9   12   2^2 * 3       {3, 9}
%e A381498 12   18   2 * 3^2       {6, 12}
%e A381498 16   30   2 * 3 * 5     {2, 4, 8, 16}
%e A381498 18   36   2^2 * 3^2     {6, 12, 18}
%e A381498 20   30   2 * 3 * 5     {10, 20}
%e A381498 24   60   2^2 * 3 * 5   {6, 12, 18, 24}
%e A381498 25   30   2 * 3 * 5     {5, 25}
%e A381498 27   39   3 * 13        {3, 9, 27}
%e A381498 28   42   2 * 3 * 7     {14, 28}
%e A381498 32   62   2 * 31        {2, 4, 8, 16, 32}
%e A381498 36   96   2^5 * 3       {6, 12, 18, 24, 36}
%t A381498 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Total@ Select[Range[n], rad[#] == r &], {n, 120}]
%o A381498 (PARI) rad(n) = factorback(factorint(n)[, 1]);
%o A381498 a(n) = my(r=rad(n)); sum(k=1, n, if(rad(k)==r, k)); \\ _Michel Marcus_, Mar 03 2025
%Y A381498 Cf. A007947, A008479, A244974, A369609.
%K A381498 nonn
%O A381498 1,2
%A A381498 _Michael De Vlieger_, Mar 03 2025