A231689 -id:A231689 - cp's OEIS frontend

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

R. Muller

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
Also the number of symbols needed to write number n in Egyptian numerals for n < 10^7. - Wojciech Graj, Jul 10 2025

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

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
Krassimir Atanassov, On Some of the Smarandache's Problems, 1999.
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

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.

a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n',r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011

[ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011

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

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 *)
DigitSum[Range[0, 100]] (* Requires v. 14 *) (* Paolo Xausa, May 17 2024 *)

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)

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

a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:

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] */

a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018

def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014

def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021

(0 to 99).map(.toString.map(.toInt - 48).sum) // Alonso del Arte, Sep 15 2019

"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"

A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

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

More terms from Hieronymus Fischer, Jun 17 2007

Edited by Michel Marcus, Nov 11 2013

A037123 a(n) = a(n-1) + sum of digits of n.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 100, 102, 105, 109, 114, 120, 127, 135, 144, 154, 165, 168, 172, 177, 183, 190, 198, 207, 217, 228, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 325, 330, 336, 343, 351, 360, 370, 381

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Sum of digits of A007908(n). - Franz Vrabec, Oct 22 2007
Also digital sum of A138793(n) for n > 0. - Bruno Berselli, May 27 2011
Sum of the digital sum of i for i from 0 to n. - N. J. A. Sloane, Nov 13 2013

N. Agronomof, Sobre una función numérica, Revista Mat. Hispano-Americana 1 (1926), 267-269.
Maurice d'Ocagne, Sur certaines sommations arithmétiques, J. Sciencias Mathematicas e Astronomicas 7 (1886), 117-128.

David A Corneth, Table of n, a(n) for n = 0..10008
P.-H. Cheo and S.-C. Yien, A problem on the k-adic representation of positive integers, Acta Math. Sinica 5, 433-438 (1955).
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
H. Delange, Sur la fonction sommatoire de la fonction "somme des chiffres", Enseignement Math. (2) 21 (1975), 31-47.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A004207, A016052, A131383, A131384, A131451.
Cf. also A074784, A231688, A231689.
Partial sums of A007953.

[ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ];  // Bruno Berselli, May 27 2011

# From N. J. A. Sloane, Nov 13 2013:
digsum:=proc(n,B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n,k,B) global digsum; local i;
add( digsum(i,B)^k,i=0..n); end;
lprint([seq(digsum(n,10),n=0..100)]); # A007953
lprint([seq(f(n,1,10),n=0..100)]); #A037123
lprint([seq(f(n,2,10),n=0..100)]); #A074784
lprint([seq(f(n,3,10),n=0..100)]); #A231688
lprint([seq(f(n,4,10),n=0..100)]); #A231689

Table[Plus@@Flatten[IntegerDigits[Range[n]]], {n, 0, 200}] (* Enrique Pérez Herrero, Oct 12 2015 *)
a[0] = 0; a[n_] := a[n - 1] + Plus @@ IntegerDigits@ n; Array[a, 70, 0] (* Robert G. Wilson v, Jul 06 2018 *)

a(n)=n*(n+1)/2-9*sum(k=1,n,sum(i=1,ceil(log(k)/log(10)),floor(k/10^i)))

a(n)={n++;my(t,i,s);c=n;while(c!=0,i++;c\=10);for(j=1,i,d=(n\10^(i-j))%10;t+=(10^(i-j)*(s*d+binomial(d,2)+d*9*(i-j)/2));s+=d);t} \\ David A. Corneth, Aug 16 2013

for $i (0..100){ @j = split "", $i; for (@j){ $sum += $; } print "$sum,"; } __END_ # gamo(AT)telecable.es

a(n) = Sum_{k=0..n} s(k) = Sum_{k=0..n} A007953(k), where s(k) denote the sum of the digits of k in decimal representation. Asymptotic expression: a(n-1) = Sum_{k=0..n-1} s(k) = 4.5*n*log_10(n) + O(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
a(n) = n*(n+1)/2 - 9*Sum_{k=1..n} Sum_{i=1..ceiling(log_10(k))} floor(k/10^i). - Benoit Cloitre, Aug 28 2003
From Hieronymus Fischer, Jul 11 2007: (Start)
G.f.: Sum_{k>=1} ((x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k)))/(1-x)^2.
a(n) = (1/2)*((n+1)*(n - 18*Sum_{k>=1} floor(n/10^k)) + 9*Sum_{k>=1} (1 + floor(n/10^k))*floor(n/10^k)*10^k).
a(n) = (1/2)*((n+1)*(2*A007953(n)-n) + 9*Sum_{k>=1} (1+floor(n/10^k))*floor(n/10^k)*10^k). (End)
a(n) = A007953(A053064(n)). - Reinhard Zumkeller, Oct 10 2008
From Wojciech Raszka, Jun 14 2019: (Start)
a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.
a(n) = a(n mod m) + MSD*a(m - 1) + (MSD*(MSD - 1)/2)*m + MSD*((n mod m) + 1), where m = 10^(A055642(n) - 1), MSD = A000030(n). (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

A074784 a(n) = a(n-1) + square of the sum of digits of n.

Original entry on oeis.org

0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 286, 290, 299, 315, 340, 376, 425, 489, 570, 670, 674, 683, 699, 724, 760, 809, 873, 954, 1054, 1175, 1184, 1200, 1225, 1261, 1310, 1374, 1455, 1555, 1676, 1820, 1836, 1861, 1897, 1946, 2010, 2091, 2191, 2312, 2456

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

a(n) = Sum_{i=0..n} digsum(i)^2, where digsum(i) = A007953(i). - N. J. A. Sloane, Nov 13 2013

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 1..990 from Indranil Ghosh)
Tom C. Brown, Powers of Digital Sums, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 207-210.
Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271; alternative link.
Robert E. Kennedy and Curtis N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 145-149.
J.-L. Mauclaire and Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
Harald Riede, Asymptotic estimation of a sum of digits, Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 72-75.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Cf. A007953, A037123, A231688, A231689, A254524.

Partial sums of A118881.

[n eq 1 select n else Self(n-1)+(&+Intseq(n))^2: n in [1..48]];  // Bruno Berselli, Jul 12 2011

Maple
```
See A037123.
```

Accumulate @ Array[(Plus @@ IntegerDigits[#])^2 &, 50] (* Amiram Eldar, Jan 20 2022 *)

a(n) = Sum_{k=1..n} s(k)^2 = Sum_{k=1..n} A007953(k)^2, where s(k) denotes the sum of the digits of k in decimal representation.
Asymptotic expression: a(n-1) = Sum_{k=1..n-1} s(k)^2 = 20.25*n*log_10(n)^2 + O(n*log_10(n)).
In general: Sum_{k=1..n-1} s(k)^m = n*((9/2)*log_10(n))^m + O(n*log_10(n)^(m-1)).

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jan 20 2022

A231688 a(n) = Sum_{i=0..n} digsum(i)^3, where digsum(i) = A007953(i).

Original entry on oeis.org

0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 2026, 2034, 2061, 2125, 2250, 2466, 2809, 3321, 4050, 5050, 5058, 5085, 5149, 5274, 5490, 5833, 6345, 7074, 8074, 9405, 9432, 9496, 9621, 9837, 10180, 10692, 11421, 12421, 13752, 15480, 15544, 15669, 15885, 16228, 16740, 17469, 18469, 19800, 21528, 23725, 23850, 24066, 24409, 24921, 25650, 26650, 27981, 29709, 31906, 34650

Offset: 0

N. J. A. Sloane, Nov 13 2013

Grabner, P. J.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F.; On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.

Indranil Ghosh, Table of n, a(n) for n = 0..10000
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
R. E. Kennedy and C. N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Q. 29, No. 2, 145-149 (1991).
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A007953, A037123, A074784, A231689.

Partial sums of A118880.

Maple
```
See A037123.
```

Accumulate[Table[Total[IntegerDigits[n]]^3,{n,0,60}]] (* Harvey P. Dale, Aug 06 2021 *)

a(n) = sum(i=0, n, sumdigits(i)^3); \\ Michel Marcus, Jan 07 2017

cp's OEIS Frontend

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

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A037123 a(n) = a(n-1) + sum of digits of n.

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A074784 a(n) = a(n-1) + square of the sum of digits of n.

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A231688 a(n) = Sum_{i=0..n} digsum(i)^3, where digsum(i) = A007953(i).

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