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A004207 a(0) = 1, a(n) = sum of digits of all previous terms.

%I A004207 M1115 #129 Jul 06 2025 11:07:53
%S A004207 1,1,2,4,8,16,23,28,38,49,62,70,77,91,101,103,107,115,122,127,137,148,
%T A004207 161,169,185,199,218,229,242,250,257,271,281,292,305,313,320,325,335,
%U A004207 346,359,376,392,406,416,427,440,448,464,478,497,517,530,538
%N A004207 a(0) = 1, a(n) = sum of digits of all previous terms.
%C A004207 If the leading 1 is omitted, this is the important sequence b(1)=1, for n >= 2, b(n) = b(n-1) + sum of digits of b(n-1). Cf. A016052, A016096, etc. - _N. J. A. Sloane_, Dec 01 2013
%C A004207 Same digital roots as A065075 (Sum of digits of the sum of the preceding numbers) and A001370 (Sum of digits of 2^n); they end in the cycle {1 2 4 8 7 5}. - _Alexandre Wajnberg_, Dec 11 2005
%C A004207 More precisely, mod 9 this sequence is 1 (1 2 4 8 7 5)*, the parenthesized part being repeated indefinitely. This shows that this sequence is disjoint from A016052. - _N. J. A. Sloane_, Oct 15 2013
%C A004207 There are infinitely many even terms (Belov 2003).
%C A004207 a(n) = A007618(n-5) for n > 57; a(n) = A006507(n-4) for n > 15. - _Reinhard Zumkeller_, Oct 14 2013
%D A004207 N. Agronomof, Problem 4421, L'Intermédiaire des mathématiciens, v. 21 (1914), p. 147.
%D A004207 D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
%D A004207 D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
%D A004207 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.
%D A004207 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A004207 G. E. Stevens and L. G. Hunsberger, A Result and a Conjecture on Digit Sum Sequences, J. Recreational Math. 27, no. 4 (1995), pp. 285-288.
%D A004207 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 37.
%H A004207 T. D. Noe, <a href="/A004207/b004207.txt">Table of n, a(n) for n = 0..10000</a>
%H A004207 A. Ya. Belov (ed.), <a href="https://www.researchgate.net/publication/299629504_Sbornik_zadac-monstrov_po_matematike">Collection of monster problems in mathematics</a> (in Russian), 2003. Problem 39.
%H A004207 D. R. Kaprekar, <a href="/A003052/a003052_2.pdf">The Mathematics of the New Self Numbers</a> [annotated and scanned]
%H A004207 J. Laroche & N. J. A. Sloane, <a href="/A004207/a004207.pdf">Correspondence, 1977</a>
%H A004207 Project Euler, <a href="https://projecteuler.net/problem=551">Problem 551: Sum of digits sequence</a>.
%H A004207 Kenneth B. Stolarsky, <a href="http://dx.doi.org/10.1090/S0002-9939-1976-0409340-X">The sum of a digitaddition series</a>, Proc. Amer. Math. Soc. 59 (1976), no. 1, 1--5. MR0409340 (53 #13099)
%H A004207 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A004207 For n>1, a(n) = a(n-1) + sum of digits of a(n-1).
%F A004207 For n > 1: a(n) = A062028(a(n-1)). - _Reinhard Zumkeller_, Oct 14 2013
%p A004207 read("transforms") :
%p A004207 A004207 := proc(n)
%p A004207     option remember;
%p A004207     if n = 0 then
%p A004207         1;
%p A004207     else
%p A004207         add( digsum(procname(i)),i=0..n-1) ;
%p A004207     end if;
%p A004207 end proc: # _R. J. Mathar_, Apr 02 2014
%p A004207 # second Maple program:
%p A004207 a:= proc(n) option remember; `if`(n<2, 1, (t->
%p A004207      t+add(i, i=convert(t, base, 10)))(a(n-1)))
%p A004207     end:
%p A004207 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jul 31 2022
%t A004207 f[s_] := Append[s, Plus @@ Flatten[IntegerDigits /@ s]]; Nest[f, {1}, 55] (* _Robert G. Wilson v_, May 26 2006 *)
%t A004207 f[n_] := n + Plus @@ IntegerDigits@n; Join[{1}, NestList[f, 1, 80]] (* _Alonso del Arte_, May 27 2006 *)
%o A004207 (Haskell)
%o A004207 a004207 n = a004207_list !! n
%o A004207 a004207_list = 1 : iterate a062028 1
%o A004207 -- _Reinhard Zumkeller_, Oct 14 2013, Sep 12 2011
%o A004207 (PARI) a(n) = { my(f(d, i) = d+vecsum(digits(d)), S=vector(n)); S[1]=1; for(k=1, n-1, S[k+1] = fold(f, S[1..k])); S } \\ _Satish Bysany_, Mar 03 2017
%o A004207 (PARI) a = 1; print1(a, ", "); for(i = 1, 50, print1(a, ", "); a = a + sumdigits(a)); \\ _Nile Nepenthe Wynar_, Feb 10 2018
%o A004207 (Python)
%o A004207 from itertools import islice
%o A004207 def agen():
%o A004207     yield 1; an = 1
%o A004207     while True: yield an; an += sum(map(int, str(an)))
%o A004207 print(list(islice(agen(), 54))) # _Michael S. Branicky_, Jul 31 2022
%Y A004207 Cf. A016052, A016096, A033298, A007612, A007953, A229527, A230107.
%Y A004207 For the base-2 analog see A010062.
%Y A004207 A065075 gives sum of digits of a(n).
%Y A004207 See A219675 for an essentially identical sequence.
%K A004207 nonn,base,easy,nice
%O A004207 0,3
%A A004207 _N. J. A. Sloane_
%E A004207 Errors from 25th term on corrected by _Leonid Broukhis_, Mar 15 1996
%E A004207 Typo in definition fixed by _Reinhard Zumkeller_, Sep 14 2011