A220849 a(n) = Product_{d|n} Product_{d_x|n , d_x <= d} d_x.

1, 2, 3, 16, 5, 432, 7, 1024, 81, 2000, 11, 71663616, 13, 5488, 10125, 1048576, 17, 816293376, 19, 2048000000, 27783, 21296, 23, 219122084616339456, 625, 35152, 59049, 15420489728, 29, 2550916800000000000, 31, 34359738368, 107811, 78608, 214375

Offset: 1

Author

Jaroslav Krizek, Dec 22 2012

Keywords

• nonn

Comments

Conjecture: sequence is injective (all terms of this sequence occur only once).

The subsequence of fixed points consists of 1 together with the primes (A008578). - Bernard Schott, Oct 26 2021

Examples

			The divisors of 6 are 1, 2, 3, 6. a(n) = 1*(1*2)*(1*2*3)*(1*2*3*6) = 1*2*6*36 = 432.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..2000 (terms 1..100 from Jaroslav Krizek)

Crossrefs

Cf. A000292, A006881, A007955, A008578 (fixed points), A064945, A266265.

Programs

Mathematica

a[n_] := Module[{d = Divisors[n], nd}, nd = Length[d]; Product[d[[i]]^(nd - i + 1), {i, 1, nd}]]; Array[a, 35] (* Amiram Eldar, Oct 23 2021 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, vecprod(select(x->(x<=d[k]), d))); \\ Michel Marcus, Oct 23 2021

Formula

a(p) = p for prime p.

From Bernard Schott, Oct 29 2021: (Start)

For p prime and k >= 0, a(p^k) = p^A000292(k).

For n = p*q, p < q primes (A006881), a(n) = p*n^3. (End)

Extensions

a(24) corrected by Seiichi Manyama, Oct 23 2021