A037123 a(n) = a(n-1) + sum of digits of n.
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
References
- 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.
Links
- 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.
Crossrefs
Programs
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Magma
[ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ]; // Bruno Berselli, May 27 2011
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Maple
# 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
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Mathematica
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 *) -
PARI
a(n)=n*(n+1)/2-9*sum(k=1,n,sum(i=1,ceil(log(k)/log(10)),floor(k/10^i)))
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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 -
Perl
for $i (0..100){ @j = split "", $i; for (@j){ $sum += $; } print "$sum,"; } __END_ # gamo(AT)telecable.es
Formula
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)
From Wojciech Raszka, Jun 14 2019: (Start)
a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.
Extensions
More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
Comments