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

1,1

Links

Table of n, a(n) for n=1..23.

Example

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.

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]<n do t:=t+1; v[t]:=add(v[k], k=t-b..t-1); od;
if v[t]=n then print(n); fi; fi; od; end: P(10^9, 1000);

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 *)

Crossrefs

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

Sequence in context: A108060 A279688 A178180 * A242185 A273527 A051810

Adjacent sequences: A269308 A269309 A269310 * A269312 A269313 A269314

Keyword

nonn,base,more

Author

Paolo P. Lava, Feb 24 2016

Extensions

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

Status

approved