cp's OEIS Frontend

A367202 If n = Product(p_i^e_i), a(n) = Sum_{i = 1..k}(rad(n)/p_i)^e_i, where rad is A007947.

%I A367202 #16 Nov 14 2023 09:17:04
%S A367202 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 A367202 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 A367202 345,22,31,1,241,1,33,52,1,18,61,1,291,26,59,1,31,1
%N A367202 If n = Product(p_i^e_i), a(n) = Sum_{i = 1..k}(rad(n)/p_i)^e_i, where rad is A007947.
%C A367202 Diverges from A028235 at a(12).
%H A367202 Michael De Vlieger, <a href="/A367202/b367202.txt">Table of n, a(n) for n = 1..10000</a>
%H A367202 Michael De Vlieger, <a href="/A367202/a367202.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%H A367202 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 A367202 For n a prime power p^k, a(n) = (p/p)^1 = 1.
%F A367202 For n a squarefree semiprime a(n) = A001414(n).
%F A367202 For p,q distinct primes a(p*q^2) = q + p^2.
%F A367202 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 A367202 a(1) = 0, the empty sum.
%e A367202 rad(6) = rad(2*3) = 6 -->a(6) = (6/2)^1 + (6/3)^1 = 3 + 2 = 5.
%e A367202 rad(12) = rad(2^2*3) = 6 -->a(12) = (6/2)^2 + (6/3)^1 = 9 + 2 = 11.
%e A367202 rad(36) = rad(2^2*3^2) = 6 --> a(36) = (6/2)^2 +(6/3)^2 = 9 + 4 = 13.
%e A367202 rad(40) = rad(2^3*5^1) = 10 -->a(40) = (10/2)^3 + (10/5)^1 = 125 + 2 = 127.
%e A367202 n = 30 = 2*3*5 a squarefree number; a(30) = (30/2) + (30/3) + (30/5) = 15 + 10 +  6 = 31
%t A367202 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 A367202 (PARI) rad(f) = factorback(f[, 1]);
%o A367202 a(n) = my(f=factor(n)); sum(i=1, #f~,(rad(f)/f[i,1])^f[i,2]); \\ _Michel Marcus_, Nov 10 2023
%Y A367202 Cf. A000040, A001414, A007947, A028235.
%K A367202 nonn
%O A367202 1,6
%A A367202 _David James Sycamore_, Nov 10 2023
%E A367202 More terms from _Michael De Vlieger_, Nov 10 2023