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URL: https://oeis.org/A004207

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A004207
a(0) = 1, a(n) = sum of digits of all previous terms.
67
1, 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440, 448, 464, 478, 497, 517, 530, 538
OFFSET
0,3
COMMENTS
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
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
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
There are infinitely many even terms (Belov 2003).
a(n) = A007618(n-5) for n > 57; a(n) = A006507(n-4) for n > 15. - Reinhard Zumkeller, Oct 14 2013
REFERENCES
N. Agronomof, Problem 4421, L'Intermédiaire des mathématiciens, v. 21 (1914), p. 147.
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 37.
LINKS
A. Ya. Belov (ed.), Collection of monster problems in mathematics (in Russian), 2003. Problem 39.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
J. Laroche & N. J. A. Sloane, Correspondence, 1977
Kenneth B. Stolarsky, The sum of a digitaddition series, Proc. Amer. Math. Soc. 59 (1976), no. 1, 1--5. MR0409340 (53 #13099)
FORMULA
For n>1, a(n) = a(n-1) + sum of digits of a(n-1).
For n > 1: a(n) = A062028(a(n-1)). - Reinhard Zumkeller, Oct 14 2013
MAPLE
read("transforms") :
A004207 := proc(n)
option remember;
if n = 0 then
1;
else
add( digsum(procname(i)), i=0..n-1) ;
end if;
end proc: # R. J. Mathar, Apr 02 2014
# Alternative:
a:= proc(n) option remember; `if`(n<2, 1, (t->
t+add(i, i=convert(t, base, 10)))(a(n-1)))
end:
seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2022
MATHEMATICA
f[s_] := Append[s, Plus @@ Flatten[IntegerDigits /@ s]]; Nest[f, {1}, 55] (* Robert G. Wilson v, May 26 2006 *)
f[n_] := n + Plus @@ IntegerDigits@n; Join[{1}, NestList[f, 1, 80]] (* Alonso del Arte, May 27 2006 *)
PROG
(Haskell)
a004207 n = a004207_list !! n
a004207_list = 1 : iterate a062028 1
-- Reinhard Zumkeller, Oct 14 2013, Sep 12 2011
(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
(PARI) a = 1; print1(a, ", "); for(i = 1, 50, print1(a, ", "); a = a + sumdigits(a)); \\ Nile Nepenthe Wynar, Feb 10 2018
(Python)
from itertools import islice
def agen():
yield 1; an = 1
while True: yield an; an += sum(map(int, str(an)))
print(list(islice(agen(), 54))) # Michael S. Branicky, Jul 31 2022
CROSSREFS
For the base-2 analog see A010062.
A065075 gives sum of digits of a(n).
See A219675 for an essentially identical sequence.
Sequence in context: A130917 A007612 A112395 * A219675 A062729 A004620
KEYWORD
nonn,base,easy,nice
EXTENSIONS
Errors from 25th term on corrected by Leonid Broukhis, Mar 15 1996
Typo in definition fixed by Reinhard Zumkeller, Sep 14 2011
STATUS
approved