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A131383
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Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').
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5
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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Eric Weisstein's World of Mathematics, Digit Sum
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FORMULA
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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 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
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EXAMPLE
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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.
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MATHEMATICA
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Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* Michael De Vlieger, Jan 03 2017 *)
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PROG
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(PARI) a(n)=sum(i=2, n+1, vecsum(digits(n, i))); \\ R. J. Cano, Jan 03 2017
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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