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

Author

Paolo P. Lava, May 04 2011

Keywords

• nonn

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).

Examples

			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.
		

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

Crossrefs

Cf. A003415, A007850,

Programs

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 *)