1,2

Sums of rows of the triangle in A138530. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 26 2008

Eric Weisstein's World of Mathematics, Digit Sum

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*ln(n))=A004125(n)+A006218(n)+O(n*ln(n)).

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

G.f.: g(x)=(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: g(x)=(1/(1-x))*(x(1+x)/(1-x)^2-sum{m>1, sum{1<j,j|m, sum{k>0,j^(1/k) is 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"). [From Alonso Delarte (alonso.delarte(AT)gmail.com), Mar 26 2009]

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.

Cf. A131384, A007953.

Sequence in context: A070881 A046669 A046670 this_sequence A139001 A090961 A073355

Adjacent sequences: A131380 A131381 A131382 this_sequence A131384 A131385 A131386

nonn

Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jul 05 2007, Jul 15 2007, Jan 07 2009