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Search: id:A065039
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| A065039 |
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If n in base 10 is d_1 d_2 ... d_k then a(n) = d_1 + d_1d_2 + d_1d_2d_3 + ... + d_1...d_k. |
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+0
4
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| 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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a(n) = sum( k>=0, floor(n/10^ k)) = n+A054899(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 03 2002
Recurrence: a(10n)=10n+a(n); a(n*10^m)=10*n*(10^m-1)/9+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(k*10^m)=k*(10^(m+1)-1)/2, 0<=k<10, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Asymptotic behavior: a(n)=10/9*n+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)<=(10n-1)/9; equality holds for powers of 10. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)>=(10n-9)/9-floor(log_10(n)); equality holds for n=10^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim inf (10n/9-a(n))=1/9, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (10n/9-log_10(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (a(n+1)-a(n)-log_10(n))=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
G.f.: g(x)=sum{k>=0, x^(10^k)/(1-x^(10^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
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EXAMPLE
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a(1234)=1370 because 1+12+123+1234=1370
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MATHEMATICA
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a[n_] := Apply[Plus, Table[FromDigits[Take[IntegerDigits[n], k]], {k, 1, Length[IntegerDigits[n]]}]]
Table[d = IntegerDigits[n]; rd = 0; While[ Length[d] > 0, rd = rd + FromDigits[d]; d = Drop[d, -1]]; rd, {n, 0, 75} ]
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CROSSREFS
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Complement of A065438. Cf. A067079, A067080, A067082.
Cf. A054899, A054861, A067080, A098844, A132027, A005187.
Adjacent sequences: A065036 A065037 A065038 this_sequence A065040 A065041 A065042
Sequence in context: A063742 A082725 A014089 this_sequence A043095 A023804 A067251
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KEYWORD
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nonn,base,nice
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AUTHOR
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Santi Spadaro (spados(AT)katamail.com), Nov 04 2001
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