A129150 The n-th arithmetic derivative of 2^3.

8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, 8592, 20096, 70464, 235072, 705280, 3023616, 13223680, 55540736, 278539264, 1392697344, 9541095424, 58609614848, 410267320320, 3397142953984, 24143851798528, 176071227916288, 1232666139967488, 9523075842834432

Offset: 0

Author

Reinhard Zumkeller, Apr 01 2007

Keywords

• nonn

Comments

Conjecture: a strictly increasing sequence. - J. Lowell, Sep 10 2008

The sequence is strictly increasing because (4*n)' = 4*n + 4*n'. - David Radcliffe, Aug 19 2014

8 is the smallest integer that has a nontrivial trajectory (not going to 0 nor reduced to a fixed point as 4) under A003415, but 15 = A090636(1) has 8 as second term in its trajectory. 20 is the next larger such integer with a distinct trajectory, but has two larger predecessors, cf. A090635. - M. F. Hasler, Nov 27 2019

In general, the trajectory of p^(p+1) under A003415 has a common factor p^p, and divided by p^p it gives the trajectory of p under A129283: n -> n + n'. Here we have the case p = 2 (see A129284 for a(n)/2^2), cf. A129151 and A129152 for p = 3 and 5. - M. F. Hasler, Nov 28 2019

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A129151, A129152, A068327, A099309, A051674, A100716, A129284.

Row n = 8 of A258651.

Programs

Haskell

a129150 n = a129150_list !! n
a129150_list = iterate a003415 8  -- Reinhard Zumkeller, Apr 29 2012

Mathematica

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; s = 2^3; Join[{s}, Table[s = dn[s], {28}]] (* T. D. Noe, Mar 07 2013 *)

PARI

A129150(n,a=8)={if(n<0, vector(-n,n, if(n>1, a=A003415(a), a)), for(n=1,n, a=A003415(a)); a)}  \\ For n<0 return the vector a[0..-n-1]. - M. F. Hasler, Nov 27 2019

Formula

a(n+1) = A003415(a(n)), a(0) = 2^3 = 8.

a(n) = A090636(n+2).

A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007

a(n) = 4*A129284(n). - M. F. Hasler, Nov 27 2019

Extensions

a(21)-a(27) from Paolo P. Lava, Apr 16 2012