A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15

Offset

0,3

Comments

Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).

Also the fixed point of the morphism 0 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -> {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - Robert G. Wilson v, Jul 27 2006

For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - Hieronymus Fischer, Jun 17 2007

It appears that a(n) is the position of 10*n in the ordered set of numbers obtained by inserting/placing one digit anywhere in the digits of n (except a zero before 1st digit). For instance, for n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where 20 is at position 2, so a(2) = 2. - Michel Marcus, Aug 01 2022

Also the total number of beads required to represent n on a Russian abacus (schoty). - P. Christopher Staecker, Mar 31 2023

a(n) / a(2n) <= 5 with equality iff n is in A169964, while a(n) / a(3n) is unbounded, since if n = (10^k + 2)/3, then a(n) = 3*k+1, a(3n) = 3, so a(n) / a(3n) = k + 1/3 -> oo when k->oo (see Diophante link). - Bernard Schott, Apr 29 2023

References

Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Krassimir Atanassov, On Some of Smarandache's Problems.

Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.

F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.

Diophante, A1762, Des chiffres à la moulinette (in French).

Ernesto Estrada and Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.

A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données (French) Acta Arith., Vol. 13 (1967/1968), pp. 259-265. MR0220693 (36 #3745)

Christian Mauduit and András Sárközy, On the arithmetic structure of sets characterized by sum of digits properties J. Number Theory, Vol. 61 , No. 1 (1996), pp. 25-38. MR1418316 (97g:11107)

Christian Mauduit and András Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith., Vol. 81, No. 2 (1997), pp. 145-173. MR1456239 (99a:11096)

Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.

Kerry Mitchell, Spirolateral image for this sequence . [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]

Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Mathematische Semesterberichte, Vol. 49 (2002), pp. 209-226.

Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme.

Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018.

Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.

Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. pp. 205-206).

Robert Walker, Self Similar Sloth Canon Number Sequences.

Eric Weisstein's World of Mathematics, Digit Sum.

Wikipedia, Digit sum.

Index entries for Colombian or self numbers and related sequences

Formula

a(A051885(n)) = n.

a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006

From Benoit Cloitre, Dec 19 2002: (Start)

a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.

a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)

From Hieronymus Fischer, Jun 17 2007: (Start)

G.f. g(x) = Sum_{k > 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).

a(n) = n - 9*Sum_{10 <= k <= n} Sum_{j|k, j >= 10} floor(log_10(j)) - floor(log_10(j-1)). (End)

From Hieronymus Fischer, Jun 25 2007: (Start)

The g.f. can be expressed in terms of a Lambert series, in that g(x) = (x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k >= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k > 1 is a power of 10, else b(k) = 0.

G.f.: g(x) = (Sum_{k > 0} (1 - 9*c(k))*x^k)/(1-x), where c(k) = Sum_{j > 1, j|k} floor(log_10(j)) - floor(log_10(j-1)).

a(n) = n - 9*Sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)

From Hieronymus Fischer, Oct 06 2007: (Start)

a(n) <= 9*(1 + floor(log_10(n)), equality holds for n = 10^m - 1, m > 0.

lim sup (a(n) - 9*log_10(n)) = 0 for n -> oo.

lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)

a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008

a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - Reinhard Zumkeller, Apr 25 2009

a(n) mod 2 = A179081(n). - Reinhard Zumkeller, Jun 28 2010

a(n) <= 9*log_10(n+1). - Vladimir Shevelev, Jun 01 2011

a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011

a(n) = Sum_{k >= 0} A031298(n, k). - Philippe Deléham, Oct 21 2011

a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - Hieronymus Fischer, Mar 24 2014

Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

Example

a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.

Maple

A007953 := proc(n) add(d, d=convert(n, base, 10)) ; end proc: # R. J. Mathar, Mar 17 2011

Mathematica

Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
Total/@IntegerDigits[Range[0, 90]] (* Harvey P. Dale, May 10 2016 *)

Prog

/* The next few PARI programs are kept for historical and pedagogical reasons.
   For practical use, the suggested and most efficient code is: A007953=sumdigits */
(PARI) a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10))) \\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n>9, n=divrem(n, 10); n[2]+a(n[1]), n)
(PARI) a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\ M. F. Hasler, Mar 22 2011
(PARI) a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:
(PARI) a(n)=sum(i=1, #n=Vecsmall(Str(n)), n[i])-48*#n \\ - M. F. Hasler, May 10 2015
/* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by M. F. Hasler, Nov 09 2018] */
(PARI) a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018
(Haskell)
a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n', r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011
(Magma) [ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011
(Smalltalk)
"Recursive version for general bases. Set base = 10 for this sequence."
digitalSum: base
| s |
base = 1 ifTrue: [^self].
(s := self // base) > 0
  ifTrue: [^(s digitalSum: base) + self - (s * base)]
  ifFalse: [^self]
"by Hieronymus Fischer, Mar 24 2014"
(Python)
def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014
(Python)
def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021
(Scala) (0 to 99).map(_.toString.map(_.toInt - 48).sum) // Alonso del Arte, Sep 15 2019
(Swift)
A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

Crossrefs

Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A169964, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).

Bisections: A004092, A004155.

For n + digsum(n) see A062028.

Sequence in context: A131650 A033930 A076314 * A080463 A209685 A114570

Adjacent sequences: A007950 A007951 A007952 * A007954 A007955 A007956

Keyword

nonn,base,nice,easy,look

Author

R. Muller

Extensions

More terms from Hieronymus Fischer, Jun 17 2007

Edited by Michel Marcus, Nov 11 2013

Status

approved