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A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.

%I A381497 #15 Jun 04 2025 10:38:37
%S A381497 0,0,0,0,0,9,0,6,6,25,0,36,0,49,45,42,0,81,0,100,84,121,0,144,45,169,
%T A381497 96,196,0,315,0,210,198,289,175,354,0,361,273,430,0,609,0,484,435,529,
%U A381497 0,648,140,655,459,676,0,801,385,826,570,841,0,1260,0,961,798
%N A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.
%C A381497 Analogous to A066760(n), the sum of row n of A133995, and A381499(n), sum of row n of A272619.
%H A381497 Michael De Vlieger, <a href="/A381497/b381497.txt">Table of n, a(n) for n = 1..16384</a>
%H A381497 Michael De Vlieger, <a href="/A381497/a381497.png">Log log scatterplot of a(n)</a>, n = 6..2^14, ignoring a(n) = 0, showing a(n) for prime power n in gold, a(n) for squarefree n in green, otherwise blue.
%F A381497 a(n) is the sum of row n of A381094.
%F A381497 a(n) = 0 for prime n and n = 4.
%F A381497 a(n) = A067392(n) - A381498(n).
%e A381497 Table of n and a(n) for select n, showing prime power decomposition of both and row n of A381094:
%e A381497    n   Factor(n) a(n)  Factor(a(n))  Row n of A381094
%e A381497   -------------------------------------------------------------------
%e A381497    6   2 * 3       9   3^2           {2,3,4}
%e A381497    8   2^3         6   2 * 3         {6}
%e A381497    9   3^2         6   2 * 3         {6}
%e A381497   10   2 * 5      25   5^2           {2,4,5,6,8}
%e A381497   12   2^2 * 3    36   2^2 * 3^2     {2,3,4,8,9,10}
%e A381497   14   2 * 7      49   7^2           {2,4,6,7,8,10,12}
%e A381497   15   3 * 5      45   3^2 * 5       {3,5,6,9,10,12}
%e A381497   16   2^4        42   2 * 3 * 7     {6,10,12,14}
%e A381497   18   2 * 3^2    81   3^4           {2,3,4,8,9,10,14,15,16}
%e A381497   20   2^2 * 5   100   2^2 * 5^2     {2,4,5,6,8,12,14,15,16,18}
%e A381497   21   3 * 7      84   2^2 * 3 * 7   {3,6,7,9,12,14,15,18}
%e A381497   22   2 * 11    121   11^2          {2,4,6,8,10,11,12,14,16,18,20}
%e A381497   24   2^3 * 3   144   2^4 * 3^2     {2,3,4,8,9,10,14,15,16,20,21,22}
%e A381497 a(6) = (2+4) + (3) = 9,
%e A381497 a(n) = 6 for n in {8, 9} since 6 is the only number less than n that shares a factor with n but does not have the same squarefree kernel as n.
%e A381497 a(10) = (2+4+6+8) + (5) = 25.
%e A381497 a(12) = (2+4+8+10) + (3+9) = 36.
%e A381497 a(14) = (2+4+6+8+10+12) + (7) = 49.
%e A381497 a(15) = (3+6+9+12) + (5+10) = 45.
%e A381497 a(16) = (6+10+12+14) = 42, etc.
%t A381497 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Total@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]
%Y A381497 Cf. A007947, A038566, A066760, A067392, A121998, A369609, A381094, A381096, A381498, A381499.
%K A381497 nonn
%O A381497 1,6
%A A381497 _Michael De Vlieger_, Mar 02 2025