A072453 Shadow transform of A000522.

0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 3, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 3, 2, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 6, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 1, 0, 0, 2, 0, 0, 1, 3

Offset: 0

Author

N. J. A. Sloane, Aug 02 2002

Keywords

• nonn
• mult
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf); see Definition 7 for the shadow transform.
• OEIS Wiki, Shadow transform.
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000522, A072456.

Programs

Maple

A000522 := proc(n)
    add(n!/k!,k=0..n) ;
end proc:
shadD := proc(a)
    local s,n ;
    s := {} ;
    for n from 0 to a-1 do
        if A000522(n) mod a = 0 then
            s := s union {n} ;
        end if;
    end do:
    s ;
end proc:
A072453 := proc(a)
    nops(shadD(a)) ;
end proc: # R. J. Mathar, Jun 24 2013
# second Maple program:
b:= proc(n) option remember; n*b(n-1)+1 end: b(0):=1:
a:= n-> add(`if`(irem(b(j), n)=0, 1, 0), j=0..n-1):
seq(a(n), n=0..150);  # Alois P. Heinz, Jun 28 2018

Mathematica

b[n_] := b[n] = n*b[n - 1] + 1 ; b[0] = 1;
a[n_] := Sum[If[Mod[b[j], n] == 0, 1, 0], {j, 0, n - 1}];
a /@ Range[0, 104] (* Jean-François Alcover, Jan 15 2020, after Alois P. Heinz *)

Extensions

More terms from Christian G. Bower, Jun 08 2005