A037123 - OEIS

%I #77 Jan 05 2025 19:51:35

%S 0,1,3,6,10,15,21,28,36,45,46,48,51,55,60,66,73,81,90,100,102,105,109,

%T 114,120,127,135,144,154,165,168,172,177,183,190,198,207,217,228,240,

%U 244,249,255,262,270,279,289,300,312,325,330,336,343,351,360,370,381

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

%C Sum of digits of A007908(n). - _Franz Vrabec_, Oct 22 2007

%C Also digital sum of A138793(n) for n > 0. - _Bruno Berselli_, May 27 2011

%C Sum of the digital sum of i for i from 0 to n. - _N. J. A. Sloane_, Nov 13 2013

%D N. Agronomof, Sobre una función numérica, Revista Mat. Hispano-Americana 1 (1926), 267-269.

%D Maurice d'Ocagne, Sur certaines sommations arithmétiques, J. Sciencias Mathematicas e Astronomicas 7 (1886), 117-128.

%H David A Corneth, <a href="/A037123/b037123.txt">Table of n, a(n) for n = 0..10008</a>

%H P.-H. Cheo and S.-C. Yien, <a href="https://actamath.cjoe.ac.cn/Jwk_sxxb_cn/EN/10.12386/A1955sxxb0033">A problem on the k-adic representation of positive integers</a>, Acta Math. Sinica 5, 433-438 (1955).

%H J. Coquet, <a href="http://dx.doi.org/10.1016/0022-314X(86)90067-3">Power sums of digital sums</a>, J. Number Theory 22 (1986), no. 2, 161-176.

%H H. Delange, <a href="http://dx.doi.org/10.5169/seals-47328">Sur la fonction sommatoire de la fonction "somme des chiffres"</a>, Enseignement Math. (2) 21 (1975), 31-47.

%H P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, <a href="https://doi.org/10.1007/978-94-011-2058-6_25">On the moments of the sum-of-digits function</a>, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.

%H Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, Preprint 2016.

%H Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="https://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.

%H J.-L. Mauclaire and Leo Murata, <a href="http://dx.doi.org/10.3792/pjaa.59.274">On q-additive functions</a>, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.

%H J.-L. Mauclaire and Leo Murata, <a href="http://dx.doi.org/10.3792/pjaa.59.441">On q-additive functions</a>, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.

%H H. Riede, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-1/riede.pdf">Asymptotic estimation of a sum of digits</a>, Fibonacci Q. 36, No. 1, 72-75 (1998).

%H J. R. Trollope, <a href="http://www.jstor.org/stable/2687954">An explicit expression for binary digital sums</a>, Math. Mag. 41 1968 21-25.

%F 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

%F 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

%F From _Hieronymus Fischer_, Jul 11 2007: (Start)

%F G.f.: Sum_{k>=1} ((x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k)))/(1-x)^2.

%F 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).

%F 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)

%F a(n) = A007953(A053064(n)). - _Reinhard Zumkeller_, Oct 10 2008

%F From _Wojciech Raszka_, Jun 14 2019: (Start)

%F a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.

%F 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)

%p # From _N. J. A. Sloane_, Nov 13 2013:

%p digsum:=proc(n,B) local a; a := convert(n, base, B):

%p add(a[i], i=1..nops(a)): end;

%p f:=proc(n,k,B) global digsum; local i;

%p add( digsum(i,B)^k,i=0..n); end;

%p lprint([seq(digsum(n,10),n=0..100)]); # A007953

%p lprint([seq(f(n,1,10),n=0..100)]); #A037123

%p lprint([seq(f(n,2,10),n=0..100)]); #A074784

%p lprint([seq(f(n,3,10),n=0..100)]); #A231688

%p lprint([seq(f(n,4,10),n=0..100)]); #A231689

%t Table[Plus@@Flatten[IntegerDigits[Range[n]]], {n, 0, 200}] (* _Enrique Pérez Herrero_, Oct 12 2015 *)

%t a[0] = 0; a[n_] := a[n - 1] + Plus @@ IntegerDigits@ n; Array[a, 70, 0] (* _Robert G. Wilson v_, Jul 06 2018 *)

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

%o (PARI) 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

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

%o (Magma) [ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ];  // _Bruno Berselli_, May 27 2011

%Y Cf. A004207, A016052, A131383, A131384, A131451.

%Y Cf. also A074784, A231688, A231689.

%Y Partial sums of A007953.

%K nonn,base,easy,changed

%O 0,3

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

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