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

%I A131383 #35 Feb 16 2025 08:33:06
%S A131383 1,3,6,8,13,16,23,25,30,35,46,46,59,66,75,74,91,91,110,112,125,136,
%T A131383 159,152,169,182,195,199,228,223,254,253,274,291,316,297,334,353,378,
%U A131383 373,414,409,452,460,475,498,545,520,557,565,598,608,661,652,693,690
%N A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').
%C A131383 Sums of rows of the triangle in A138530. - _Reinhard Zumkeller_, Mar 26 2008
%H A131383 Hieronymus Fischer, <a href="/A131383/b131383.txt">Table of n, a(n) for n = 1..10000</a>
%H A131383 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>
%F A131383 a(n) = n^2-sum{k>0, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.
%F A131383 a(n) = n^2-sum{2<=p<=n, (p-1)*sum{0<k<=log_p(n), floor(n/p^k)}}.
%F A131383 a(n) = n^2-A024916(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.
%F A131383 a(n) = A004125(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.
%F A131383 Asymptotic behavior: a(n) = (1-Pi^2/12)*n^2 + O(n*log(n)) = A004125(n) + A006218(n) + O(n*log(n)).
%F A131383 Lim a(n)/n^2 = 1 - Pi^2/12 for n-->oo.
%F A131383 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))}= }).
%F A131383 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}).
%F A131383 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)))}}}.
%F A131383 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).
%F A131383 a(n) = sum{1<=p<=n, ds_p(n)} where ds_p = digital sum base p.
%F A131383 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
%e A131383 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.
%t A131383 Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* _Michael De Vlieger_, Jan 03 2017 *)
%o A131383 (PARI) a(n)=sum(i=2,n+1,vecsum(digits(n,i))); \\ _R. J. Cano_, Jan 03 2017
%Y A131383 Cf. A131384, A007953.
%K A131383 nonn,base
%O A131383 1,2
%A A131383 _Hieronymus Fischer_, Jul 05 2007, Jul 15 2007, Jan 07 2009