A190014 a(0)=0, a(1)=1, if n = (n-1)', then a(n)=0 otherwise a(n)=2*a((n-1)'), where n’ is the arithmetic derivative of n.

0, 1, 0, 2, 2, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 8, 2, 4, 8, 16, 2, 4, 8, 16, 32, 4, 2, 0, 2, 4, 4, 16, 4, 4, 2, 8, 8, 4, 2, 8, 2, 4, 16, 8, 2, 32, 4, 8, 4, 16, 2, 512, 8, 8, 16, 0, 2, 8, 2, 8, 16, 4, 4, 16, 2, 4, 16, 0, 2, 16, 2, 16, 1024, 4, 4, 0, 2, 256, 4, 16, 2, 8, 16, 8, 4, 4, 2, 32, 4, 8, 8, 64, 4, 4, 2, 8, 32, 4

Offset

0,4

Comments

Only power of 2 and zeros. If p is prime than a(p+1)=2.

If n’>n+1 than a(n+1) is not immediately available. It is necessary to find a(n’)=2*a((n’-1)’) and, if necessary, to repeat the process until a term can be calculate. For instance:

a(9)=2*a(12) -> a(12)=2 and therefore a(9)=4.

Again: a(55)=2*a(81) -> a(81)=2*a(176) -> a(176)=8*a(112) -> a(112)=16 and therefore a(176)=128 -> a(81)=256 -> a(55)=512.

First zero at a(31)=2*a(31) and for all Giuga numbers plus one (31, 859, 1723, 66199, etc.). This because the so far known Giuga numbers satisfy the equation n’=n+1. Other zeros for a(59)=8*a(31), a(71)=16*a(31), a(79)=32*a(31), a(106)=32*a(31), etc.

The general equation a(n+1)=k*a(n’), with k integer and |k|>1, a(0)=0, a(1)=1, leads to the following sequence: 0, 1, 0, k, k, k^2, k, k^3,k, k^2, k^2, k^4, k, k^3, k, k^3, k^2, k^2, k, k^3, etc.

For k=1 or k=-1 we get and incongruence because of a(31)=a(31).

Links

Paolo P. Lava, Table of n, a(n) for n = 0..5000

Paolo P. Lava, Plot of the first 5000 terms of the sequence

Example

a(0)=0
a(1)=1
a(2)=a(1+1)=2*a(1')=2*a(0)=0
a(3)=a(2+1)=2*a(2')=2*a(1)=2
a(4)=a(3+1)=2*a(3')=2*a(1)=2
a(5)=a(4+1)=2*a(4')=2*a(4)=4
a(6)=a(5+1)=2*a(5')=2*a(1)=2
a(7)=a(6+1)=2*a(6')=2*a(5)=8  etc.

Maple

with(numtheory);
P:=proc(i)
local a, f, n, p, pfs, t;
a:=array(0..100000); a[0]:=0; a[1]:=1; t:=2; lprint(0, a[0]); lprint(1, a[1]);
for n from 1 by 1 to i do
    pfs:=ifactors(n)[2]; f:=n*add(op(2, p)/op(1, p), p=pfs);
    a[n+1]:=t*a[f]; lprint(n+1, a[n+1]);
od;
end:

Mathematica

dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Module[{d = dn[n - 1]}, If[d == n, 0, 2 a[d]]]; Array[a, 100, 0] (* T. D. Noe, May 05 2011 *)

Crossrefs

Cf. A003415, A007850,

Sequence in context: A324604 A270366 A072478 * A353214 A350062 A100577

Adjacent sequences: A190011 A190012 A190013 * A190015 A190016 A190017

Keyword

nonn

Author

Paolo P. Lava, May 04 2011

Status

approved