|
%I #280 Jun 18 2023 11:41:19
%S 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,
%T 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,
%U 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
%N Digital sum (i.e., sum of digits) of n; also called digsum(n).
%C Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).
%C 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
%C For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - _Hieronymus Fischer_, Jun 17 2007
%C It appears that a(n) is the position of 10*n in the ordered set of numbers obtained by inserting/placing one digit anywhere in the digits of n (except a zero before 1st digit). For instance, for n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where 20 is at position 2, so a(2) = 2. - _Michel Marcus_, Aug 01 2022
%C Also the total number of beads required to represent n on a Russian abacus (schoty). - _P. Christopher Staecker_, Mar 31 2023
%C a(n) / a(2n) <= 5 with equality iff n is in A169964, while a(n) / a(3n) is unbounded, since if n = (10^k + 2)/3, then a(n) = 3*k+1, a(3n) = 3, so a(n) / a(3n) = k + 1/3 -> oo when k->oo (see Diophante link). - _Bernard Schott_, Apr 29 2023
%D Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
%H N. J. A. Sloane, <a href="/A007953/b007953.txt">Table of n, a(n) for n = 0..10000</a>
%H Krassimir Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of Smarandache's Problems</a>.
%H Jean-Luc Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.
%H F. M. Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Dekking/dek25.html">The Thue-Morse Sequence in Base 3/2</a>, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/5453-a1762-des-chiffres-a-la-moulinette">A1762, Des chiffres à la moulinette</a> (in French).
%H Ernesto Estrada and Puri Pereira-Ramos, <a href="https://doi.org/10.1155/2018/9893867">Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts</a>, Complexity, Vol. 2018 (2018), Article ID 9893867.
%H A. O. Gel'fond, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa13115.pdf">Sur les nombres qui ont des propriétés additives et multiplicatives données</a> (French) Acta Arith., Vol. 13 (1967/1968), pp. 259-265. MR0220693 (36 #3745)
%H Christian Mauduit and András Sárközy, <a href="http://dx.doi.org/10.1006/jnth.1998.2229">On the arithmetic structure of sets characterized by sum of digits properties</a> J. Number Theory, Vol. 61 , No. 1 (1996), pp. 25-38. MR1418316 (97g:11107)
%H Christian Mauduit and András Sárközy, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa81/aa8122.pdf"> On the arithmetic structure of the integers whose sum of digits is fixed</a>, Acta Arith., Vol. 81, No. 2 (1997), pp. 145-173. MR1456239 (99a:11096)
%H Kerry Mitchell, <a href="http://kerrymitchellart.com/articles/Spirolateral-Type_Images_from_Integer_Sequences.pdf">Spirolateral-Type Images from Integer Sequences</a>, 2013.
%H Kerry Mitchell, <a href="/A007953/a007953.jpg">Spirolateral image for this sequence</a> . [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]
%H Jan-Christoph Puchta and Jürgen Spilker, <a href="http://dx.doi.org/10.1007/s00591-002-0048-4">Altes und Neues zur Quersumme</a>, Mathematische Semesterberichte, Vol. 49 (2002), pp. 209-226.
%H Jan-Christoph Puchta and Jürgen Spilker, <a href="http://www.math.uni-rostock.de/~schlage-puchta/papers/Quersumme.pdf">Altes und Neues zur Quersumme</a>.
%H Maxwell Schneider and Robert Schneider, <a href="https://arxiv.org/abs/1807.06710">Digit sums and generating functions</a>, arXiv:1807.06710 [math.NT], 2018.
%H Jeffrey O. Shallit, <a href="http://www.jstor.org/stable/2322179">Problem 6450</a>, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; <a href="http://www.jstor.org/stable/2322523">Two series, solution to Problem 6450</a>, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
%H Vladimir Shevelev, <a href="http://journals.impan.pl/aa/Inf/126-3-1.html">Compact integers and factorials</a>, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. pp. 205-206).
%H Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Digit_sum">Digit sum</a>.
%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F a(A051885(n)) = n.
%F a(n) <= 9(log_10(n)+1). - _Stefan Steinerberger_, Mar 24 2006
%F From _Benoit Cloitre_, Dec 19 2002: (Start)
%F a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.
%F a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)
%F From _Hieronymus Fischer_, Jun 17 2007: (Start)
%F G.f. g(x) = Sum_{k > 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).
%F a(n) = n - 9*Sum_{10 <= k <= n} Sum_{j|k, j >= 10} floor(log_10(j)) - floor(log_10(j-1)). (End)
%F From _Hieronymus Fischer_, Jun 25 2007: (Start)
%F 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.
%F 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)).
%F a(n) = n - 9*Sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
%F From _Hieronymus Fischer_, Oct 06 2007: (Start)
%F a(n) <= 9*(1 + floor(log_10(n)), equality holds for n = 10^m - 1, m > 0.
%F lim sup (a(n) - 9*log_10(n)) = 0 for n -> oo.
%F lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)
%F a(n) = A138530(n, 10) for n > 9. - _Reinhard Zumkeller_, Mar 26 2008
%F a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - _Reinhard Zumkeller_, Apr 25 2009
%F a(n) mod 2 = A179081(n). - _Reinhard Zumkeller_, Jun 28 2010
%F a(n) <= 9*log_10(n+1). - _Vladimir Shevelev_, Jun 01 2011
%F a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - _Alexander R. Povolotsky_, Oct 09 2011
%F a(n) = Sum_{k >= 0} A031298(n, k). - _Philippe Deléham_, Oct 21 2011
%F a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - _Hieronymus Fischer_, Mar 24 2014
%F Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - _Amiram Eldar_, Jun 03 2021
%e a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.
%p A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # _R. J. Mathar_, Mar 17 2011
%t Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* _Stefan Steinerberger_, Mar 24 2006 *)
%t Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
%t Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* _Robert G. Wilson v_, Jul 27 2006 *)
%t Total/@IntegerDigits[Range[0,90]] (* _Harvey P. Dale_, May 10 2016 *)
%o /* The next few PARI programs are kept for historical and pedagogical reasons.
%o For practical use, the suggested and most efficient code is: A007953=sumdigits */
%o (PARI) a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10))) \\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n>9, n=divrem(n, 10); n[2]+a(n[1]), n)
%o (PARI) a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\ _M. F. Hasler_, Mar 22 2011
%o (PARI) a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:
%o (PARI) a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\ - _M. F. Hasler_, May 10 2015
%o /* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by _M. F. Hasler_, Nov 09 2018] */
%o (PARI) a(n) = sumdigits(n); \\ _Altug Alkan_, Apr 19 2018
%o (Haskell)
%o a007953 n | n < 10 = n
%o | otherwise = a007953 n' + r where (n',r) = divMod n 10
%o -- _Reinhard Zumkeller_, Nov 04 2011, Mar 19 2011
%o (Magma) [ &+Intseq(n): n in [0..87] ]; // _Bruno Berselli_, May 26 2011
%o (Smalltalk)
%o "Recursive version for general bases. Set base = 10 for this sequence."
%o digitalSum: base
%o | s |
%o base = 1 ifTrue: [^self].
%o (s := self // base) > 0
%o ifTrue: [^(s digitalSum: base) + self - (s * base)]
%o ifFalse: [^self]
%o "by _Hieronymus Fischer_, Mar 24 2014"
%o (Python)
%o def A007953(n):
%o return sum(int(d) for d in str(n)) # _Chai Wah Wu_, Sep 03 2014
%o (Python)
%o def a(n): return sum(map(int, str(n))) # _Michael S. Branicky_, May 22 2021
%o (Scala) (0 to 99).map(_.toString.map(_.toInt - 48).sum) // _Alonso del Arte_, Sep 15 2019
%o (Swift)
%o A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // _Egor Khmara_, Jun 15 2021
%Y Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A169964, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).
%Y Bisections: A004092, A004155.
%Y For n + digsum(n) see A062028.
%K nonn,base,nice,easy,look
%O 0,3
%A R. Muller
%E More terms from _Hieronymus Fischer_, Jun 17 2007
%E Edited by _Michel Marcus_, Nov 11 2013
|
