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A269312 Consider 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 the arithmetic derivative of x.

%I A269312 #14 May 22 2019 14:19:32
%S A269312 14,51,145,285,629,708,807,1318,2362,2548,2869,3789,4087,4811,6031,
%T A269312 6355,10201,15563,17143,17287,17561,19883,20567,21731,22429,23461,
%U A269312 26269,27301,30967,33389,69529,73211,85927,86087,90133,96781,110159,116011,159767,161701,162055,190079
%N A269312 Consider 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 the arithmetic derivative of x.
%H A269312 Lars Blomberg, <a href="/A269312/b269312.txt">Table of n, a(n) for n = 1..663</a>
%e A269312 14’ = 9 : 1 + 4 = 5; 4 + 5 = 9.
%e A269312 51’ = 20 : 5 + 1 = 6; 1 + 6 = 7; 6 + 7  = 13; 7 + 13 = 20.
%p A269312 with(numtheory): P:=proc(q,h) local a,b,c,k,n,p,t,v; v:=array(1..h);
%p A269312 for n from 1 to q do a:=n; b:=ilog10(a)+1; if b>1 then
%p A269312 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);c:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p A269312 while v[t]<c do t:=t+1; v[t]:=add(v[k], k=t-b..t-1); od;
%p A269312 if v[t]=c then print(n); fi; fi; od; end: P(10^9,1000);
%t A269312 dn[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@n]]]; (* after Michael Somos,Apr 12 2011 *)
%t A269312 Select[Range[10^5], # >= 10 && (s = dn[#]; d = IntegerDigits[#]; While[Total[d] < s, d = Join[Rest[d], {Total[d]}]]; Total[d] == s) &] (* _Robert Price_, May 22 2019 *)
%Y A269312 Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269311.
%K A269312 nonn,base
%O A269312 1,1
%A A269312 _Paolo P. Lava_, Feb 24 2016