cp's OEIS Frontend

A189553 Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415).

%I A189553 #15 Mar 30 2012 17:22:58
%S A189553 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 A189553 57,85,121,20,95,119,143,46,69,133,169,27,115,187,161,209,221,30,58,
%U A189553 16,28,87,247,62,93,145,253,289,155,203,299,323,217,361,45,74
%N A189553 Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415).
%C A189553 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 A189553 See A003415.
%H A189553 T. D. Noe, <a href="/A189553/b189553.txt">Rows n = 4..1000, flattened</a>
%H A189553 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 A189553 The triangle begins
%e A189553 4
%e A189553 6
%e A189553 9
%e A189553 10
%e A189553 15
%e A189553 14
%e A189553 21, 25
%e A189553 none
%e A189553 8, 35
%e A189553 22
%e A189553 33, 49
%e A189553 26
%e A189553 12, 39, 55
%t A189553 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 A189553 Cf. A003415, A098699, A099302, A099303.
%K A189553 nonn,tabf
%O A189553 4,1
%A A189553 _T. D. Noe_, Apr 23 2011