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A269311 Consider the arithmetic derivative of a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.

%I A269311 #15 May 22 2019 21:01:50
%S A269311 8,18,21,160,1590,2420,18620,69229,83790,279964,494520,598810,676450,
%T A269311 1183147,4233720,5600348,14217074,20025836,64278677,425208387,
%U A269311 604048830,750851470,1981942354
%N A269311 Consider the arithmetic derivative of a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.
%e A269311 8’ = 12 : 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8.
%e A269311 18’ = 21 : 2 + 1 = 3; 1 + 3 = 4; 3 + 4 = 7; 4 + 7 = 11; 7 + 11 = 18.
%p A269311 with(numtheory): P:=proc(q,h) local a,b,k,p,t,v; global n; v:=array(1..h);
%p A269311 for n from 1 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=ilog10(a)+1; if b>1 then
%p A269311 for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
%p A269311 while v[t]<n do t:=t+1; v[t]:=add(v[k], k=t-b..t-1); od;
%p A269311 if v[t]=n then print(n); fi; fi; od; end: P(10^9, 1000);
%t A269311 dn[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@n]]]; (* from _Michael Somos_, Apr 12 2011 *)
%t A269311 Select[Range[10^5], dn[#] >= 10 && (d = IntegerDigits[dn[#]]; While[Total[d] < #, d = Join[Rest[d], {Total[d]}]]; Total[d] == #) &] (* _Robert Price_, May 22 2019 *)
%Y A269311 Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269312.
%K A269311 nonn,base,more
%O A269311 1,1
%A A269311 _Paolo P. Lava_, Feb 24 2016
%E A269311 a(19)-a(23) from _Lars Blomberg_, Jan 18 2018