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A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.

%I A381499 #11 Jun 04 2025 10:39:01
%S A381499 0,0,0,0,0,0,0,6,6,6,0,10,0,28,28,42,0,39,0,65,65,80,0,102,45,126,96,
%T A381499 159,0,111,0,210,148,210,138,253,0,280,221,338,0,342,0,411,366,444,0,
%U A381499 547,140,563,403,601,0,700,344,708,512,750,0,751,0,868,703,930
%N A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.
%C A381499 Analogous to A066760(n), the sum of row n of A133995, and A381497(n), sum of row n of A381094.
%H A381499 Michael De Vlieger, <a href="/A381499/b381499.txt">Table of n, a(n) for n = 1..16384</a>
%H A381499 Michael De Vlieger, <a href="/A381499/a381499.png">Log log scatterplot of a(n)</a>, n = 8..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 A381499 a(n) is the sum of row n of A272619.
%F A381499 a(n) = 0 for prime n, n = 4, and n = 6.
%e A381499 Table of n and a(n) for select n, showing prime power decomposition of the latter and row n of A272619:
%e A381499  n   a(n)  Factor(a(n))  Row n of A272619
%e A381499 -----------------------------------------------------
%e A381499  8     6   2 * 3         {6}
%e A381499  9     6   2 * 3         {6}
%e A381499 10     6   2 * 3         {6}
%e A381499 12    10   2 * 5         {10}
%e A381499 14    28   2^2 * 7       {6,10,12}
%e A381499 15    28   2^2 * 7       {6,10,12}
%e A381499 16    42   2 * 3 * 7     {6,10,12,14}
%e A381499 18    39   3 * 13        {10,14,15}
%e A381499 20    65   5 * 13        {6,12,14,15,18}
%e A381499 21    65   5 * 13        {6,12,14,15,18}
%e A381499 22    80   2^4 * 5       {6,10,12,14,18,20}
%e A381499 24   102   2 * 3 * 17    {10,14,15,20,21,22}
%e A381499 25    45   3^2 * 5       {10,15,20}
%e A381499 26   126   2 * 3^2 * 7   {6,10,12,14,18,20,22,24}
%e A381499 27    96   2^5 * 3       {6,12,15,18,21,24}
%e A381499 28   159   3 * 53        {6,10,12,18,20,21,22,24,26}
%t A381499 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; If[PrimeQ[n], 0, Total@ Select[Range[n], And[1 < GCD[#, n] < #, ! Divisible[n, rad[#]]] &]], {n, 120}]
%Y A381499 Cf. A007947, A038566, A066760, A121998, A243823, A272619, A381497.
%K A381499 nonn
%O A381499 1,8
%A A381499 _Michael De Vlieger_, Mar 02 2025