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A378689 a(n) = product of divisors d of n that are not coreful.

%I A378689 #26 Feb 07 2025 22:51:52
%S A378689 1,1,1,1,1,6,1,1,1,10,1,24,1,14,15,1,1,54,1,40,21,22,1,192,1,26,1,56,
%T A378689 1,27000,1,1,33,34,35,216,1,38,39,320,1,74088,1,88,135,46,1,3072,1,
%U A378689 250,51,104,1,1458,55,448,57,58,1,25920000,1,62,189,1,65,287496
%N A378689 a(n) = product of divisors d of n that are not coreful.
%H A378689 Michael De Vlieger, <a href="/A378689/b378689.txt">Table of n, a(n) for n = 1..10000</a>
%H A378689 Michael De Vlieger, <a href="/A378689/a378689.png">Log log scatterplot of log_10(a(n))</a> for n = 1..2^20.
%H A378689 Michael De Vlieger, <a href="/A378689/a378689_1.png">Log log scatterplot of log_10(a(n))</a> for n = 1..2^16 (ignoring a(n) = 1, i.e., n that is a power of a prime), showing a(n) such that n is in A286708 in purple, n in A332785 in blue, n in A120944 in green, highlighting n in A002110 in large green points.
%F A378689 a(n) = A007955(n) / A308360(n).
%F A378689 a(n) = 1 for powers of primes n (i.e., n in A000961), since d | n such that d > 1 are coreful.
%e A378689 Table of n, a(n), and divisors that are not coreful that produce a(n) for select n:
%e A378689    n     a(n)
%e A378689   -----------------------------
%e A378689    1       1   (empty product)
%e A378689    2       1 = 1
%e A378689    3       1 = 1
%e A378689    4       1 = 1
%e A378689    5       1 = 1
%e A378689    6       6 = 1*2*3
%e A378689   10      10 = 1*2*5
%e A378689   12      24 = 1*2*3*4
%e A378689   14      14 = 1*2*7
%e A378689   15      15 = 1*3*5
%e A378689   18      54 = 1*2*3*9
%e A378689   20      40 = 1*2*4*5
%e A378689   21      21 = 1*3*7
%e A378689   22      22 = 1*2*11
%e A378689   24     192 = 1*2*3*4*8
%e A378689   30   27000 = 1*2*3*5*6*10*15
%e A378689   36     216 = 1*2*3*4*9
%t A378689 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Times @@ Select[Divisors[n], rad[#] != r &], {n, 120}]
%o A378689 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A378689 a(n) = my(d=divisors(n), c=rad(n), p=1); for (i=1, #d~, if (rad(d[i]) != c, p *= d[i])); p; \\ _Michel Marcus_, Feb 07 2025
%Y A378689 Cf. A007955, A027750, A308135 (sums), A308360 (product of coreful divisors of n).
%K A378689 nonn,easy
%O A378689 1,6
%A A378689 _Michael De Vlieger_, Feb 05 2025