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.

8, 18, 21, 160, 1590, 2420, 18620, 69229, 83790, 279964, 494520, 598810, 676450, 1183147, 4233720, 5600348, 14217074, 20025836, 64278677, 425208387, 604048830, 750851470, 1981942354

Offset: 1

Author

Paolo P. Lava, Feb 24 2016

Keywords

• nonn
• base
• more

Examples

			8’ = 12 : 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8.
18’ = 21 : 2 + 1 = 3; 1 + 3 = 4; 3 + 4 = 7; 4 + 7 = 11; 7 + 11 = 18.
		

Crossrefs

Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269312.

Programs

Maple

with(numtheory): P:=proc(q,h) local a,b,k,p,t,v; global n; v:=array(1..h);
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
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);
while v[t]
				

Mathematica

dn[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@n]]]; (* from Michael Somos, Apr 12 2011 *)
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 *)

Extensions

a(19)-a(23) from Lars Blomberg, Jan 18 2018