A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

1, 3, 6, 8, 13, 16, 23, 25, 30, 35, 46, 46, 59, 66, 75, 74, 91, 91, 110, 112, 125, 136, 159, 152, 169, 182, 195, 199, 228, 223, 254, 253, 274, 291, 316, 297, 334, 353, 378, 373, 414, 409, 452, 460, 475, 498, 545, 520, 557, 565, 598, 608, 661, 652, 693, 690

Offset

1,2

Comments

Sums of rows of the triangle in A138530. - Reinhard Zumkeller, Mar 26 2008

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Digit Sum

Formula

a(n) = n^2-sum{k>0, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = n^2-sum{2<=p<=n, (p-1)*sum{0<k<=log_p(n), floor(n/p^k)}}.

a(n) = n^2-A024916(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = A004125(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

Asymptotic behavior: a(n) = (1-Pi^2/12)*n^2 + O(n*log(n)) = A004125(n) + A006218(n) + O(n*log(n)).

Lim a(n)/n^2 = 1 - Pi^2/12 for n-->oo.

G.f.: (1/(1-x))*(x(1+x)/(1-x)^2-sum{k>0,sum{j>1,(j-1)*x^(j^k)/(1-x^(j^k))}= }).

Also: (1/(1-x))*(x(1+x)/(1-x)^2-sum{m>1, sum{1<j,j|m, sum{k>0,j^(1/k) is an integer, j^(1/k)-1}}*x^m}).

a(n) = n^2-sum{1<m<=n,sum{k>0,sum{1<j,j|m, (j^(1/k)-1)(floor(j^(1/k))-floor((j-1)^(1/k)))}}}.

Recurrence: a(n)=a(n-1)-b(n)+2n-1, where b(n)=sum{1<j,j|n, sum{1<=k<=log_2(j),fract(j^(1/k))=0, j^(1/k)-1}} and fract(x)=fractional part of x=x-floor(x).

a(n) = sum{1<=p<=n, ds_p(n)} where ds_p = digital sum base p.

a(n) = A043306(n) + n (that sequence ignores unary) = A014837(n) + n + 1 (that sequence ignores unary and base n in which n is "10"). - Alonso del Arte, Mar 26 2009

Example

5 = 11111(base 1) = 101(base 2) = 12(base 3) = 11(base 4) = 10(base 5). Thus a(5) = ds_1(5)+ds_2(5)+ds_3(5)+ds_4(5)+ds_5(5) = 5+2+3+2+1 = 13.

Mathematica

Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* Michael De Vlieger, Jan 03 2017 *)

Prog

(PARI) a(n)=sum(i=2, n+1, vecsum(digits(n, i))); \\ R. J. Cano, Jan 03 2017

Crossrefs

Cf. A131384, A007953.

Sequence in context: A046669 A352773 A046670 * A219730 A373083 A337484

Adjacent sequences: A131380 A131381 A131382 * A131384 A131385 A131386

Keyword

nonn,base

Author

Hieronymus Fischer, Jul 05 2007, Jul 15 2007, Jan 07 2009

Status

approved