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

2,3

Links

Table of n, a(n) for n=2..25.

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)

Maple

P:= proc(p) local a, b, m, n, i, ok, pd, t1, t2, t3; a:=0; pd:=1;
for n from 2 by 1 to p do b:=1000000000039; ok:=0; if n<=1 then a:=0; ok:=1; fi; if isprime(n) then a:=1; ok:=1; fi; if ok=0 then t1:=ifactor(b*n); m:=nops(t1); t2:=0; for i from 1 to m do t3:=op(i, t1); if nops(t3)=1 then t2:=t2+1/op(t3); else t2:=t2+op(2, t3)/op(op(1, t3)); fi; od;
t2:=t2-1/b; a:=n*t2; fi; pd:=pd*a; print(pd); od; end: P(100);
# Alternative program 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 *)

Crossrefs

Cf. A003415, A190144, A209873.

Sequence in context: A261568 A087213 A117857 * A180967 A231884 A052923

Adjacent sequences: A165556 A165557 A165558 * A165560 A165561 A165562

Keyword

easy,nonn

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 22 2009

Extensions

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

Status

approved