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Search: id:A131383
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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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+0
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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Sums of rows of the triangle in A138530. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 26 2008
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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)=sum{1<=k<=n, ds_p(k)} where ds_p = digital sum base p.
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).
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CROSSREFS
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Cf. A131384.
Cf. A131384, A007953.
Sequence in context: A070881 A046669 A046670 this_sequence A139001 A090961 A073355
Adjacent sequences: A131380 A131381 A131382 this_sequence A131384 A131385 A131386
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jul 05 2007, Jul 15 2007
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