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A007953
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Digital sum (i.e. sum of digits) of n.
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433
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15
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OFFSET
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0,3
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COMMENTS
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Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).
Also the fixed point of the morphism 0 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -> {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - Robert G. Wilson v, Jul 27 2006.
For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - Hieronymus Fischer, Jun 17 2007
a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008
a(A058369(n)) =A 004159(A058369(n)); a(A000290(n)) = A004159(n). [From Reinhard Zumkeller, Apr 25 2009]
a(n) mod 2 = A179081(n). [From Reinhard Zumkeller, Jun 28 2010]
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REFERENCES
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K. Atanassov, On the 16-th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
J.-C. Puchta, J. Spilker, Altes und Neues zur Quersumme, Math. Semesterber, 49 (2002), 209-226.
V. Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no.3,195-236 (cf. pp.205-206).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..10000
K. Atanassov, On Some of Smarandache's Problems
Robert Walker, Self Similar Sloth Canon Number Sequences
Eric Weisstein's World of Mathematics, Digit Sum
Wikipedia, Digit sum
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FORMULA
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a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006
a(0) = 0, a(10n+i) = a(n) + i 0 <= i <= 9; a(n) = n - 9*(sum(k > 0, floor(n/10^k)) = n - 9*A054899(n). - Benoit Cloitre, Dec 19, 2002
From Hieronymus Fischer, Jun 17 2007: (Start)
G.f. g(x) = sum{k > 0, (x^k-x^(k+10^k)-9x^(10^k))/(1-x^(10^k))}/(1-x).
a(n) = n - 9*sum{10 <= k <= n, sum{j|k, j >= 10, floor(log_10(j))-floor(log_10(j-1))}}. (End)
From Hieronymus Fischer, Jun 25 2007: (Start)
The g.f. can be expressed in terms of a Lambert series, in that g(x)=(x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k >= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k > 1 is a power of 10, else b(k) = 0.
G.f.: g(x) = sum{k > 0, (1-9*c(k))*x^k}/(1-x), where c(k) = sum{j > 1, j|k, floor(log_10(j)) - floor(log_10(j-1))}.
a(n) = n - 9*sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
From Hieronymus Fischer, Oct 06 2007: (Start)
a(n) <= 9*(1+floor(log_10(n)), equality holds for n = 10^m - 1, m > 0.
lim sup (a(n) - 9*log_10(n)) = 0 for n --> infinity.
lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n --> infinity. (End)
a(A051885(n)) = n.
a(n) <= 9*log_10(n+1). - Vladimir Shevelev, Jun 01 2011.
a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011.
a(n) = Sum_k >= 0 {A031298(n, k)}. - Philippe Deléham, Oct 21 2011.
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EXAMPLE
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a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.
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MAPLE
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A007953 := proc(n) add(d, d=convert(n, base, 10)) ; end proc: # R. J. Mathar, Mar 17 2011
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MATHEMATICA
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Table[Sum[DigitCount[n][[i]]*i, {i, 1, 9}], {n, 1, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits@n, {n, 0, 87}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 9}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
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PROG
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(PARI) a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10)))
(PARI) A007953(n, b=10)={ local(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1] }
(Haskell)
a007953 n | n < 10 = n
| otherwise = a007953 n' + r where (n', r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011
(MAGMA) [ &+Intseq(n): n in [0..87] ]; // Bruno Berselli, May 26 2011
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CROSSREFS
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Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A007953, A003132, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A179082-A179085, A108971, A179987, A179988, A180018, A180019.
Sequence in context: A131650 A033930 A076314 * A080463 A209685 A114570
Adjacent sequences: A007950 A007951 A007952 * A007954 A007955 A007956
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KEYWORD
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nonn,base,nice,easy
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AUTHOR
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R. Muller
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EXTENSIONS
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More terms from Hieronymus Fischer, Jun 17 2007
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STATUS
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approved
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