A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

1, 2, 3, 8, 5, 1, 7, 1, 27, 1, 11, 48, 13, 1, 1, 16, 17, 162, 19, 80, 1, 1, 23, 16, 125, 1, 1, 112, 29, 25, 31, 512, 1, 1, 1, 1944, 37, 1, 1, 25, 41, 49, 43, 176, 405, 1, 47, 48, 343, 1250, 1, 208, 53, 324, 1, 49, 1, 1, 59, 9, 61, 1, 567, 8, 1, 121, 67, 272, 1, 49, 71, 9, 73, 1

Offset: 1

Author

Reinhard Zumkeller, May 12 2003

Keywords

• nonn

Comments

a(n) = r(n,tau(n)), where r is defined as follows:

let d(n,j) = j-th divisor of n, 1 <= j <= tau(n) = A000005(n), r(n,1)=d(n,1), r(n,j) = if d(n,j) divides r(n,j-1) then r(n,j-1)/d(n,j) else r(n,j-1)*d(n,j), 1 < j <= tau(n);

p prime: a(p)=p, a(p^2)=p^3, a(p^3)=1, a(p^k)=p^A008344(k+1);

a(m)=1 iff m multiplicatively perfect: a(A007422(k))=1.

a(A084111(n)) = A084111(n). - Reinhard Zumkeller, Jul 31 2014

Examples

			Divisors of 48 = {1,2,3,4,6,8,12,16,24,48}: 1*2*3 = 6 -> 6*4 = 24 -> 24/6 = 4 -> 4*8 = 32 -> 32*12 = 384 -> 384/16 = 24 -> 24/24 = 1 -> 1*48 = a(48);
divisors of 49 = {1,7,49}: 1*7 = 7 -> 7*49 = 343 = a(49);
divisors of 50 = {1,2,5,10,25,50}: 1*2*5 = 10 -> 10/10 = 1 -> 1*25 = 25 -> 25*50 = 1250 = a(50).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor Product.
• Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Crossrefs

Cf. A027750, A084111 (fixed points), A084113, A084114.

Programs

Haskell

a084110 = foldl (/*) 1 . a027750_row where
   x /* y = if m == 0 then x' else x*y where (x',m) = divMod x y
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

Mathematica

a[n_] := Module[{d = Divisors[n], e}, e[i_] := e[i] = If[i == 1, 1, If[Divisible[e[i-1], d[[i]]], e[i-1]/d[[i]], e[i-1] d[[i]]]]; e[Length[d]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 10 2021 *)

Extensions

Corrected and extended by David Wasserman, Dec 14 2004