%I #16 Nov 14 2023 09:17:04
%S 0,1,1,1,1,5,1,1,1,7,1,11,1,9,8,1,1,7,1,27,10,13,1,29,1,15,1,51,1,31,
%T 1,1,14,19,12,13,1,21,16,127,1,41,1,123,28,25,1,83,1,9,20,171,1,11,16,
%U 345,22,31,1,241,1,33,52,1,18,61,1,291,26,59,1,31,1
%N If n = Product(p_i^e_i), a(n) = Sum_{i = 1..k}(rad(n)/p_i)^e_i, where rad is A007947.
%C Diverges from A028235 at a(12).
%H Michael De Vlieger, <a href="/A367202/b367202.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A367202/a367202.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%H Michael De Vlieger, <a href="/A367202/a367202_1.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, highlighting squareful numbers that are not prime powers in large light blue.
%F For n a prime power p^k, a(n) = (p/p)^1 = 1.
%F For n a squarefree semiprime a(n) = A001414(n).
%F For p,q distinct primes a(p*q^2) = q + p^2.
%F For n a squarefree number with prime divisors p_1,p_2..p_k, a(n) = Sum_{i = 1..k}(n/p_i) see Example
%e a(1) = 0, the empty sum.
%e rad(6) = rad(2*3) = 6 -->a(6) = (6/2)^1 + (6/3)^1 = 3 + 2 = 5.
%e rad(12) = rad(2^2*3) = 6 -->a(12) = (6/2)^2 + (6/3)^1 = 9 + 2 = 11.
%e rad(36) = rad(2^2*3^2) = 6 --> a(36) = (6/2)^2 +(6/3)^2 = 9 + 4 = 13.
%e rad(40) = rad(2^3*5^1) = 10 -->a(40) = (10/2)^3 + (10/5)^1 = 125 + 2 = 127.
%e n = 30 = 2*3*5 a squarefree number; a(30) = (30/2) + (30/3) + (30/5) = 15 + 10 + 6 = 31
%t Array[Function[{r, w}, Total[Power @@@ Transpose@ {r/w[[All, 1]], w[[All, -1]]}]] @@ {Times @@ #[[All, 1]], #} &@ FactorInteger[#] &, 120] (* _Michael De Vlieger_, Nov 10 2023 *)
%o (PARI) rad(f) = factorback(f[, 1]);
%o a(n) = my(f=factor(n)); sum(i=1, #f~,(rad(f)/f[i,1])^f[i,2]); \\ _Michel Marcus_, Nov 10 2023
%Y Cf. A000040, A001414, A007947, A028235.
%K nonn
%O 1,6
%A _David James Sycamore_, Nov 10 2023
%E More terms from _Michael De Vlieger_, Nov 10 2023