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Numbers n such that x' = n has two solutions, where x' is the arithmetic derivative (A003415) of x.
2

%I #11 Mar 30 2012 17:22:58

%S 10,12,14,18,20,21,28,31,38,39,45,55,61,71,81,87,101,103,111,119,123,

%T 129,131,147,183,185,199,211,213,215,241,243,255,269,291,297,299,327,

%U 339,343,351,355,359,361,363,381,395,399,401,411,421,433,439,471,493

%N Numbers n such that x' = n has two solutions, where x' is the arithmetic derivative (A003415) of x.

%C Ufnarovski and Ahlander conjecture that this sequence, and any such sequence that has numbers n such that x' = n has k solutions, is infinite. See A098700 and A189481 for the k=0 and 1 cases. It appears that the only even terms here are 10, 12, 14, 18, 20, 28, and 38. The prime terms are in A189556.

%D See A003415.

%H T. D. Noe, <a href="/A189555/b189555.txt">Table of n, a(n) for n = 1..1000</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.

%F n such that A099302(n) = 2.

%Y Cf. A003415, A098700 (no solution), A099302, A189481 (1 solution).

%K nonn

%O 1,1

%A _T. D. Noe_, Apr 24 2011