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A004207 a(0) = 1, a(n) = sum of digits of all previous terms.
(Formerly M1115)
62

%I M1115

%S 1,1,2,4,8,16,23,28,38,49,62,70,77,91,101,103,107,115,122,127,137,148,

%T 161,169,185,199,218,229,242,250,257,271,281,292,305,313,320,325,335,

%U 346,359,376,392,406,416,427,440,448,464,478,497,517,530,538

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

%C 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 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 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 There are infinitely many even terms (Belov 2003).

%C a(n) = A007618(n-5) for n > 57; a(n) = A006507(n-4) for n > 15. - _Reinhard Zumkeller_, Oct 14 2013

%D N. Agronomof, Problem 4421, L'Intermédiaire des mathématiciens, v. 21 (1914), p. 147.

%D D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.

%D D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.

%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D 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.

%H T. D. Noe, <a href="/A004207/b004207.txt">Table of n, a(n) for n = 0..10000</a>

%H A. Ya. Belov (ed.), <a href="http://bookre.org/reader?file=579337&amp;pg=0">Collection of monster problems in mathematics</a> (in Russian), 2003. Problem 39.

%H D. R. Kaprekar, <a href="/A003052/a003052.pdf">The Mathematics of the New Self Numbers</a> [annotated and scanned]

%H J. Laroche & N. J. A. Sloane, <a href="/A004207/a004207.pdf">Correspondence, 1977</a>

%H Project Euler, <a href="https://projecteuler.net/problem=551">Problem 551: Sum of digits sequence</a>.

%H 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 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>

%F For n>1, a(n) = a(n-1) + sum of digits of a(n-1).

%F For n > 1: a(n) = A062028(a(n-1)). - _Reinhard Zumkeller_, Oct 14 2013

%p read("transforms") :

%p A004207 := proc(n)

%p option remember;

%p if n = 0 then

%p 1;

%p else

%p add( digsum(procname(i)),i=0..n-1) ;

%p end if;

%p end proc: # _R. J. Mathar_, Apr 02 2014

%t f[s_] := Append[s, Plus @@ Flatten[IntegerDigits /@ s]]; Nest[f, {1}, 55] (* _Robert G. Wilson v_, May 26 2006 *)

%t f[n_] := n + Plus @@ IntegerDigits@n; Join[{1}, NestList[f, 1, 80]] (* _Alonso del Arte_, May 27 2006 *)

%o (Haskell)

%o a004207 n = a004207_list !! n

%o a004207_list = 1 : iterate a062028 1

%o -- _Reinhard Zumkeller_, Oct 14 2013, Sep 12 2011

%o (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 (PARI) a = 1; print1(a, ", "); for(i = 1, 50, print1(a, ", "); a = a + sumdigits(a)); \\ _Nile Nepenthe Wynar_, Feb 10 2018

%Y Cf. A016052, A016096, A033298, A007612, A007953, A229527, A230107.

%Y For the base-2 analog see A010062.

%Y A065075 gives sum of digits of a(n).

%Y See A219675 for an essentially identical sequence.

%K nonn,base,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E Errors from 25th term on corrected by _Leonid Broukhis_, Mar 15 1996

%E Typo in definition fixed by _Reinhard Zumkeller_, Sep 14 2011

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Last modified November 22 03:20 EST 2019. Contains 329383 sequences. (Running on oeis4.)