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%I #15 Mar 30 2012 17:22:58
%S 4,6,9,10,15,14,21,25,8,35,22,33,49,26,12,39,55,65,77,34,51,91,18,38,
%T 57,85,121,20,95,119,143,46,69,133,169,27,115,187,161,209,221,30,58,
%U 16,28,87,247,62,93,145,253,289,155,203,299,323,217,361,45,74
%N Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415).
%C Row 0 contains 0 and 1. Row 1 contains all primes. Rows 2 and 3 are empty. Hence, we start this table with row 4. The length of row n is A099302(n). The first term in row n is A098699(n). The last term is A099303(n). Row n is the set I(n) in the paper by Ufnarovski and Ahlander. They show that all terms in row n are <= (n/2)^2. The upper bound is attained when n = 2p, where p is a prime.
%D See A003415.
%H T. D. Noe, <a href="/A189553/b189553.txt">Rows n = 4..1000, flattened</a>
%H Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%e The triangle begins
%e 4
%e 6
%e 9
%e 10
%e 15
%e 14
%e 21, 25
%e none
%e 8, 35
%e 22
%e 33, 49
%e 26
%e 12, 39, 55
%t dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; nn = 100; d = Array[dn, (nn/2)^2]; Table[Flatten[Position[d, n]], {n, 4, nn}]
%Y Cf. A003415, A098699, A099302, A099303.
%K nonn,tabf
%O 4,1
%A _T. D. Noe_, Apr 23 2011
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