A168386 Arithmetic derivative of the double factorial of n.

0, 0, 1, 1, 12, 8, 112, 71, 1472, 1269, 17408, 14904, 270336, 204147, 4199424, 4143285, 87834624, 72462870, 1797783552, 1411253955, 40414740480, 36183623805, 937430876160, 845972658090, 26095323709440, 24311657884500, 707908274749440, 869872809558375

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..800

Formula

a(n) = A003415(A006882(n)). - R. J. Mathar, Nov 26 2009

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 proc:
A168386 := proc(n) A003415(doublefactorial(n)) ; end proc:
seq(A168386(n), n=0..80) ; # R. J. Mathar, Nov 26 2009
# second Maple program:
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember;
      `if`(n<2, 0, a(n-2)*n+doublefactorial(n-2)*d(n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 06 2015

Mathematica

d[n_] := n*Total[#2/#1& @@@ FactorInteger[n]];
a[0] = a[1] = 0; a[n_] := d[n!!];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 18 2018 *)

Prog

(Python 3.8+)
from collections import Counter
from sympy import factorial2, factorint
def A168386(n): return sum((factorial2(n)*e//p for p, e in sum((Counter(factorint(m)) for m in range(n, 1, -2)), start=Counter({2:0})).items())) if n > 1 else 0 # Chai Wah Wu, Jun 12 2022

Crossrefs

Cf. A003415, A068311.

Sequence in context: A206478 A164675 A121961 * A338825 A338809 A038334

Adjacent sequences: A168383 A168384 A168385 * A168387 A168388 A168389

Keyword

easy,nonn

Author

Paolo P. Lava and Giorgio Balzarotti, Nov 24 2009

Extensions

Program replaced by a structured program - R. J. Mathar, Nov 26 2009

Status

approved