A004427 -id:A004427 - cp's OEIS frontend

A178403 Numbers containing the rounded up arithmetic mean of their digits at least once, cf. A004427.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 133, 134, 135, 143, 145, 146, 147, 153, 154, 157, 158, 159, 164, 169, 174, 175, 185

Offset: 1

Reinhard Zumkeller, May 27 2010

A178401(a(n)) > 0; complement of A178402.
A010785, A050278, A178358, A178359 are subsequences;
a(n) = A131207(n) for n < 48;
a(n) = A134336(n) for n < 48;
a(n+1) = A032981(n) for n < 38.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A178401 Number of times the rounded up arithmetic mean of digits of n occurs in n, cf. A004427.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 1, 0, 0

Offset: 0

Reinhard Zumkeller, May 27 2010

a(A178402(n)) = 0; a(A050278(n)) = 1; a(A178403(n)) > 0;

a(A010785(n)) > 0; a(A178358(n)) > 0; a(A178359(n)) > 0.

R. Zumkeller, Table of n, a(n) for n = 0..25000

A178358 Rounded up arithmetic mean of digits of n prepended to n, cf. A004427.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111, 212, 213, 314, 315, 416, 417, 518, 519, 120, 221, 222, 323, 324, 425, 426, 527, 528, 629, 230, 231, 332, 333, 434, 435, 536, 537, 638, 639, 240, 341, 342, 443, 444, 545, 546, 647, 648, 749, 350, 351, 452, 453

Offset: 0

Reinhard Zumkeller, May 27 2010

A000030(a(n)) = A004427(a(n)) = A004427(n);
a(A178359(n)) = A178359(a(n));
subsequence of A178403.

			n=8379 --> A004427(n) = ceiling((8+3+7+9)/4) = 7
--> a(8379) = 7*10^4 + 8379 = 78379.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

a178358 n = read $ show (a004427 n) ++ show n :: Integer
-- Reinhard Zumkeller, Mar 17 2014

Table[Ceiling[Mean[IntegerDigits[n]]]*10^IntegerLength[n]+n,{n,0,100}] (* Harvey P. Dale, Apr 21 2019 *)

a(n) = A004427(n)*A011557(A055642(n)) + n.

A178359 Rounded up arithmetic mean of digits of n appended to n, cf. A004427.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 122, 132, 143, 153, 164, 174, 185, 195, 201, 212, 222, 233, 243, 254, 264, 275, 285, 296, 302, 312, 323, 333, 344, 354, 365, 375, 386, 396, 402, 413, 423, 434, 444, 455, 465, 476, 486, 497, 503, 513, 524, 534

Offset: 0

Reinhard Zumkeller, May 27 2010

A010879(a(n)) = A004427(a(n)) = A004427(n);
a(A178358(n)) = A178358(a(n));
subsequence of A178403.

			For n=8379: A004427(n) = ceiling((8+3+7+9)/4) = 7; so a(8379) = 10*8379 + 7 = 83797.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

amd[n_]:=Module[{m=Ceiling[Mean[IntegerDigits[n]]]},n*10^IntegerLength[ m]+ m]; Array[amd,60,0] (* Harvey P. Dale, Aug 12 2015 *)

a(n) = A008592(n) + A004427(n).

A178402 Numbers not containing the rounded up arithmetic mean of their digits, cf. A004427.

Original entry on oeis.org

13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 30, 31, 35, 36, 37, 38, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 64, 68, 69, 70, 71, 72, 73, 74, 75, 79, 80, 81, 82, 83, 84, 85, 86, 90, 91, 92, 93, 94, 95, 96, 97, 103, 104, 105, 106

Offset: 1

Reinhard Zumkeller, May 27 2010

A178401(a(n)) = 0; complement of A178403.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

Original entry on oeis.org

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

Offset: 0

R. Muller

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
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
Also the total number of beads required to represent n on a Russian abacus (schoty). - P. Christopher Staecker, Mar 31 2023
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
Also the number of symbols needed to write number n in Egyptian numerals for n < 10^7. - Wojciech Graj, Jul 10 2025

			a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
Krassimir Atanassov, On Some of the Smarandache's Problems, 1999.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Diophante, A1762, Des chiffres à la moulinette (in French).
Ernesto Estrada and Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.
A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données (French) Acta Arith., Vol. 13 (1967/1968), pp. 259-265. MR0220693 (36 #3745)
Christian Mauduit and András Sárközy, On the arithmetic structure of sets characterized by sum of digits properties J. Number Theory, Vol. 61, No. 1 (1996), pp. 25-38. MR1418316 (97g:11107)
Christian Mauduit and András Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith., Vol. 81, No. 2 (1997), pp. 145-173. MR1456239 (99a:11096)
Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.
Kerry Mitchell, Spirolateral image for this sequence . [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Mathematische Semesterberichte, Vol. 49 (2002), pp. 209-226.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme.
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. pp. 205-206).
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.
Wikipedia, Digit sum.
Index entries for Colombian or self numbers and related sequences

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).
Bisections: A004092, A004155.
For n + digsum(n) see A062028.

a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n',r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011

[ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Mar 17 2011

Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
Total/@IntegerDigits[Range[0,90]] (* Harvey P. Dale, May 10 2016 *)
DigitSum[Range[0, 100]] (* Requires v. 14 *) (* Paolo Xausa, May 17 2024 *)

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)

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

a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:

a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\ M. F. Hasler, May 10 2015
/* 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] */

a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018

def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014

def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021

(0 to 99).map(.toString.map(.toInt - 48).sum) // Alonso del Arte, Sep 15 2019

"Recursive version for general bases. Set base = 10 for this sequence."
digitalSum: base
| s |
base = 1 ifTrue: [^self].
(s := self // base) > 0
  ifTrue: [^(s digitalSum: base) + self - (s * base)]
  ifFalse: [^self]
"by Hieronymus Fischer, Mar 24 2014"

A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

a(A051885(n)) = n.
a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.
a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)
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 -> oo.
lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)
a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008
a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - Reinhard Zumkeller, Apr 25 2009
a(n) mod 2 = A179081(n). - Reinhard Zumkeller, Jun 28 2010
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
a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - Hieronymus Fischer, Mar 24 2014
Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

More terms from Hieronymus Fischer, Jun 17 2007

Edited by Michel Marcus, Nov 11 2013

A061383 Arithmetic mean of digits is an integer.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129

Offset: 0

Amarnath Murthy, May 03 2001

A004426(a(n)) = A004427(a(n)). - Reinhard Zumkeller, May 27 2010
A175688 is a subsequence; complement of A180157; A180160(a(n))=0. - Reinhard Zumkeller, Aug 15 2010
It seems "obvious" that n log n << a(n) < n log n; is this true? - Charles R Greathouse IV, Feb 06 2013

			123 is a term as the arithmetic mean is (1+2+3)/3 = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007953, A055642.
Cf. A004426, A004427.
Cf. A175688, A180157, A180160.

a061383 n = a061383_list !! (n-1)
a061383_list = filter (\x -> mod (a007953 x) (a055642 x) == 0) [0..]
-- Reinhard Zumkeller, Jun 18 2013

[0] cat [n: n in [1..130] | IsZero(&+Intseq(n) mod #Intseq(n))];  // Bruno Berselli, Jun 30 2011

[0] cat [n: n in [1..130] | IsIntegral(&+Intseq(n)/#Intseq(n))];   // Bruno Berselli, Feb 09 2016

Select[Range[0,129],IntegerQ[Total[x=IntegerDigits[#]]/Length[x]] &] (* Jayanta Basu, May 17 2013 *)
Select[Range[0,200],IntegerQ[Mean[IntegerDigits[#]]]&] (* Harvey P. Dale, Dec 31 2022 *)

is(n)=my(v=digits(n));sum(i=1,#v,v[i])%#v==0 \\ Charles R Greathouse IV, Feb 06 2013

def ok(n): return n == 0 or sum(d:=list(map(int, str(n))))%len(d) == 0
print([k for k in range(130) if ok(k)]) # Michael S. Branicky, Apr 23 2025

A004426 Arithmetic mean of digits of n (rounded down).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 0, 0, 1, 1, 1, 2, 2, 2

Offset: 0

N. J. A. Sloane

From Reinhard Zumkeller, May 27 2010: (Start)
A004427(n) <= a(n);
a(A061383(n)) = A004427(A061383(n));
a(A000040(n)) = A074461(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A175688.

a004426 n = (a007953 n) `div` (a055642 n)
-- Reinhard Zumkeller, Jun 18 2013

Table[Floor[Mean[IntegerDigits[n]]], {n, 0, 99}] (* Enrique Pérez Herrero, Sep 28 2013 *)

a(n) = floor(A007953(n)/A055642(n)). - Reinhard Zumkeller, May 27 2010

A178361 Numbers with rounded up arithmetic mean of digits = 1.

Original entry on oeis.org

1, 10, 11, 20, 100, 101, 102, 110, 111, 120, 200, 201, 210, 300, 1000, 1001, 1002, 1003, 1010, 1011, 1012, 1020, 1021, 1030, 1100, 1101, 1102, 1110, 1111, 1120, 1200, 1201, 1210, 1300, 2000, 2001, 2002, 2010, 2011, 2020, 2100, 2101, 2110, 2200, 3000, 3001

Offset: 1

Reinhard Zumkeller, May 27 2010

A004427(a(n)) = 1;

A000027 = union of A178362, A178363, A178364, A178365, A178366, A178367, A178368, A178369, and this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A055642.

a178361 n = a178361_list !! (n-1)
a178361_list = [x | x <- [1..], a007953 x <= a055642 x]
-- Reinhard Zumkeller, May 06 2015

Select[Range[4000],Ceiling[Mean[IntegerDigits[#]]]==1&] (* Harvey P. Dale, Dec 31 2019 *)

A007953(a(n)) <= A055642(a(n)). - Reinhard Zumkeller, May 06 2015

A178363 Numbers with rounded up arithmetic mean of digits = 3.

Original entry on oeis.org

3, 14, 15, 23, 24, 32, 33, 41, 42, 50, 51, 60, 106, 107, 108, 115, 116, 117, 124, 125, 126, 133, 134, 135, 142, 143, 144, 151, 152, 153, 160, 161, 162, 170, 171, 180, 205, 206, 207, 214, 215, 216, 223, 224, 225, 232, 233, 234, 241, 242, 243, 250, 251, 252, 260

Offset: 1

Reinhard Zumkeller, May 27 2010

A004427(a(n)) = 3;

A000027 = union of A178361, A178362, A178364, A178365, A178366, A178367, A178368, A178369, and this sequence.

R. Zumkeller, Table of n, a(n) for n = 1..1000

cp's OEIS Frontend

A178403 Numbers containing the rounded up arithmetic mean of their digits at least once, cf. A004427.

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A178401 Number of times the rounded up arithmetic mean of digits of n occurs in n, cf. A004427.

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A178358 Rounded up arithmetic mean of digits of n prepended to n, cf. A004427.

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A178359 Rounded up arithmetic mean of digits of n appended to n, cf. A004427.

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A178402 Numbers not containing the rounded up arithmetic mean of their digits, cf. A004427.

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A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

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A061383 Arithmetic mean of digits is an integer.

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