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.

14, 51, 145, 285, 629, 708, 807, 1318, 2362, 2548, 2869, 3789, 4087, 4811, 6031, 6355, 10201, 15563, 17143, 17287, 17561, 19883, 20567, 21731, 22429, 23461, 26269, 27301, 30967, 33389, 69529, 73211, 85927, 86087, 90133, 96781, 110159, 116011, 159767, 161701, 162055, 190079

Offset

1,1

Links

Lars Blomberg, Table of n, a(n) for n = 1..663

Example

14’ = 9 : 1 + 4 = 5; 4 + 5 = 9.
51’ = 20 : 5 + 1 = 6; 1 + 6 = 7; 6 + 7  = 13; 7 + 13 = 20.

Maple

with(numtheory): P:=proc(q, h) local a, b, c, k, n, p, t, v; v:=array(1..h);
for n from 1 to q do a:=n; 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); c:=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
while v[t]<c do t:=t+1; v[t]:=add(v[k], k=t-b..t-1); od;
if v[t]=c 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]]]; (* after Michael Somos, Apr 12 2011 *)
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 *)

Crossrefs

Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269311.

Sequence in context: A009961 A059997 A007588 * A129025 A113907 A125740

Adjacent sequences: A269309 A269310 A269311 * A269313 A269314 A269315

Keyword

nonn,base

Author

Paolo P. Lava, Feb 24 2016

Status

approved