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A335665 Product of the refactorable divisors of n.

%I A335665 #40 Feb 16 2025 08:34:00
%S A335665 1,2,1,2,1,2,1,16,9,2,1,24,1,2,1,16,1,324,1,2,1,2,1,4608,1,2,9,2,1,2,
%T A335665 1,16,1,2,1,139968,1,2,1,640,1,2,1,2,9,2,1,4608,1,2,1,2,1,324,1,896,1,
%U A335665 2,1,1440,1,2,9,16,1,2,1,2,1,2,1,1934917632,1,2,1,2,1,2,1,51200
%N A335665 Product of the refactorable divisors of n.
%H A335665 Antti Karttunen, <a href="/A335665/b335665.txt">Table of n, a(n) for n = 1..10000</a>
%H A335665 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RefactorableNumber.html">Refactorable Number</a>
%F A335665 a(n) = Product_{d|n} d^c(d), where c(n) is the refactorable characteristic of n (A336040).
%F A335665 a(n) = Product_{d|n} d^(1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).
%F A335665 a(p) = 1 for odd primes p. - _Wesley Ivan Hurt_, Nov 28 2021
%e A335665 a(6) = 2; The divisors of 6 are {1,2,3,6}. 1 and 2 are refactorable since d(1) = 1|1 and d(2) = 2|2, so a(6) = 1 * 2 = 2.
%e A335665 a(7) = 1; The divisors of 7 are {1,7} and 1 is the only refactorable divisor of 7. So a(7) = 1.
%e A335665 a(8) = 16; The divisors of 8 are {1,2,4,8}. 1, 2 and 8 are refactorable since d(1) = 1|1, d(2) = 2|2 and d(8) = 4|8, so a(8) = 1 * 2 * 8 = 16.
%e A335665 a(9) = 9; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9, so a(9) = 1 * 9 = 9.
%t A335665 a[n_] := Product[If[Divisible[d, DivisorSigma[0, d]], d, 1], {d, Divisors[n]}]; Array[a, 60] (* _Amiram Eldar_, Nov 24 2021 *)
%o A335665 (PARI) isr(n) = n%numdiv(n)==0; \\ A033950
%o A335665 a(n) = my(d=divisors(n)); prod(k=1, #d, if (isr(d[k]), d[k], 1)); \\ _Michel Marcus_, Jul 18 2020
%Y A335665 Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A336041 (number of refactorable divisors), A335182 (their sum).
%Y A335665 Cf. also A349322 (similar formula, but with sum instead of product).
%K A335665 nonn,easy
%O A335665 1,2
%A A335665 _Wesley Ivan Hurt_, Jul 17 2020