A165559 Product of the arithmetic derivatives from 2 to n.

1, 1, 4, 4, 20, 20, 240, 1440, 10080, 10080, 161280, 161280, 1451520, 11612160, 371589120, 371589120, 7803371520, 7803371520, 187280916480, 1872809164800, 24346519142400, 24346519142400, 1071246842265600, 10712468422656000

Offset: 2

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 22 2009

Keywords

• easy
• nonn

Crossrefs

Cf. A003415, A190144, A209873.

Programs

Maple

A003415 := proc(n) local pfs ; if n <= 1 then 0 ; else pfs := ifactors(n)[2] ; n*add(op(2,p)/op(1,p),p=pfs) ; fi; end:
A165559 := proc(n) mul( A003415(k),k=2..n) ; end: seq( A165559(n),n=2..30) ; # R. J. Mathar, Sep 26 2009

Mathematica

d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; a[n_] := Product[d[k], {k, 2, n}]; Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Feb 21 2014 *)

Formula

a(n) = Product_{k=2..n} A003415(k).

From Amiram Eldar, Nov 15 2020: (Start)

Sum_{n>=2} 1/a(n) = A190144.

Sum_{n>=2} (-1)^n/a(n) = A209873. (End)

Extensions

Entries checked by R. J. Mathar, Sep 26 2009