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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; a:=1; t:=2; lprint(0, a); lprint(1, a); for n from 1 by 1 to i do     pfs:=ifactors(n); 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; dn = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[]/f[])]]; a = 0; a = 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 * A100577 A328710 A018818 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 September 16 15:58 EDT 2021. Contains 347473 sequences. (Running on oeis4.)