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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A132801 A270366 A072478 * A100577 A018818 A157019

Adjacent sequences:  A190011 A190012 A190013 * A190015 A190016 A190017

KEYWORD

nonn

AUTHOR

Paolo P. Lava, May 04 2011

STATUS

approved

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Last modified February 22 17:51 EST 2018. Contains 299469 sequences. (Running on oeis4.)