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%I #9 Aug 27 2015 08:17:18
%S 16,108,3294172,7975979
%N Numbers n such that n+(n+1) = n'+(n+1)', where n' is the arithmetic derivative of n.
%C a(5) > 10^13. - _Giovanni Resta_, Mar 13 2014
%C a(5) > 5*10^13. - _Hiroaki Yamanouchi_, Aug 27 2015
%e 16' = 32, 17' = 1 and 16 + 17 = 32 + 1 = 33.
%e 108' = 216, 109' = 1 and 108 + 109 = 216 + 1 = 217.
%e 3294172' = 6588344, 3274173' = 1 and 3294172 + 3294173 = 6588344 + 1 = 6588345.
%e 7975979' = 941063, 7975980' = 15010896 and 7975979 + 7975980 = 941063 + 15010896 = 15951959.
%p with(numtheory); P:= proc(q) local a,b,n,p; for n from 1 to q do
%p a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p b:=(n+1)*add(op(2,p)/op(1,p),p=ifactors(n+1)[2]);
%p if 2*n+1=a+b then print(n); fi; od; end: P(10^9);
%Y Cf. A003415.
%K nonn,more
%O 1,1
%A _Paolo P. Lava_, Feb 19 2014