A217928 -id:A217928 - cp's OEIS frontend

A227378 Smallest number with n = sum of distinct digits in decimal representation, cf. A217928.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 189, 289, 389, 489, 589, 689, 789, 1789, 2789, 3789, 4789, 5789, 6789, 16789, 26789, 36789, 46789, 56789, 156789, 256789, 356789, 456789, 1456789, 2456789, 3456789, 13456789, 23456789, 123456789

Offset: 0

Reinhard Zumkeller, Jul 09 2013

A217928(a(n)) = A007953(a(n)) = n and A217928(m) < n for m < a(n).

import Data.List (elemIndex); import Data.Maybe (fromJust)
a227378 = fromJust . (`elemIndex` a217928_list)

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

A216407 Sum of decimal digits not appearing in n.

Original entry on oeis.org

45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 44, 44, 42, 41, 40, 39, 38, 37, 36, 35, 43, 42, 43, 40, 39, 38, 37, 36, 35, 34, 42, 41, 40, 42, 38, 37, 36, 35, 34, 33, 41, 40, 39, 38, 41, 36, 35, 34, 33, 32, 40, 39, 38, 37, 36, 40, 34, 33, 32, 31, 39, 38, 37, 36, 35, 34, 39, 32, 31, 30, 38, 37, 36, 35, 34, 33, 32, 38, 30, 29, 37, 36, 35, 34, 33, 32, 31, 30, 37, 28, 36, 35, 34, 33, 32, 31, 30, 29, 28, 36, 44

Offset: 0

Xenia Sheinerman, Oct 15 2012

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

a216407 = (45 -) . a217928  -- Reinhard Zumkeller, Jul 09 2013

Total[Complement[Range[0,9],IntegerDigits[#]]]&/@Range[0,100] (* Harvey P. Dale, Aug 13 2013 *)

{ a(n) = local( d = vecsort( eval(Vec(Str(n))),,8) ); 45 - sum(i=1,#d,d[i]) }

a(n) = 45 - A217928(n).

a(n) = 0 for almost all n. Average order is n^-0.045 where the exponent is log(0.9)/log(10). - Charles R Greathouse IV, Oct 15 2012

A280911 Numbers n such that sum of decimal digits of n equals number of prime divisors of n counted with multiplicity and sum of distinct decimal digits of n equals number of distinct primes dividing n.

Original entry on oeis.org

30, 102, 1002, 1012, 1210, 2001, 2120, 3010, 10002, 10030, 20001, 20112, 20120, 100012, 100030, 101020, 102010, 110020, 110120, 120001, 121120, 200001, 200120, 211100, 221120, 230010, 300010, 320320, 400010, 400140, 1000002, 1000012, 1000140, 1000230, 1001020, 1003002, 1004010, 1010120, 1011300, 1013310, 1021100

Offset: 1

Ilya Gutkovskiy, Jan 10 2017

Numbers n such that A007953(n) = A001222(n) and A217928(n) = A001221(n).

			20112 is in the sequence because 20112 = 2^4*3*419  (6 prime factors, 3 distinct), 2 + 0 + 1 + 1 + 2 = 6 and 2 + 0 + 1 = 3.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Sum
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A001222, A007953, A050689, A050690, A057531, A217928.

Select[Range[1100000], Total[IntegerDigits[#1]] == PrimeOmega[#1] && Total[Union[IntegerDigits[#1]]] == PrimeNu[#1] &]

A356981 Numbers k such that the sum of distinct digits of k equals the sum of the prime divisors of k.

Original entry on oeis.org

2, 3, 5, 7, 84, 144, 160, 250, 343, 468, 735, 936, 975, 1125, 1215, 1375, 1408, 1600, 1694, 1872, 2401, 2500, 2646, 2880, 3920, 4913, 6084, 6318, 6860, 7296, 7695, 8624, 8704, 8788, 9126, 10125, 10240, 10816, 11264, 12672, 12675, 14641, 14896, 16000

Offset: 1

Tanya Khovanova, Sep 09 2022

Similar to A070275, where distinctness of digits is not required.

			144 = 2^4*3^2 and 1+4=2+3. Thus, 144 is in this sequence.

Cf. A070275, A217928.

Select[Range[2, 20000],Total[Union[IntegerDigits[#]]] ==  Total[Transpose[FactorInteger[#]][[1]]] &]

isok(k) = vecsum(Set(digits(k))) == vecsum(factor(k)[, 1]); \\ Michel Marcus, Sep 12 2022

from itertools import count, islice
from sympy import primefactors
def A356981_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:sum(int(d) for d in set(str(k)))==sum(primefactors(k)), count(max(startvalue,1)))
A356981_list = list(islice(A356981_gen(),30)) # Chai Wah Wu, Sep 12 2022

A357263 Numbers k such that the sum of the distinct digits of k is equal to the product of the prime divisors of k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 24, 343, 375, 392, 640, 686, 2401, 3375, 4802, 4913, 6400, 13122, 14336, 14641, 30375, 33614, 64000, 468750, 640000, 1703936, 2725888, 2839714, 2883584, 4687500, 5537792, 6298560, 6400000, 7864320, 13668750, 14172488, 19267584, 21807104, 26040609, 28629151

Offset: 1

Alexandru Petrescu, Sep 21 2022

64*10^k is a term of the sequence for every positive integer k.

			375 = 3*5^3. 3+7+5 = 3*5. Thus 375 is a term.

Cf. A008472, A217928.

Select[Range[10^6], Plus @@ Union[IntegerDigits[#]] == Times @@ FactorInteger[#][[;;,1]] &] (* Amiram Eldar, Sep 21 2022 *)

isok(k) = vecsum(Set(digits(k))) == vecprod(factor(k)[, 1]);

from math import prod
from sympy import primefactors
def ok(n): return n and sum(map(int, set(str(n)))) == prod(primefactors(n))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Sep 22 2022

More terms from Michel Marcus, Sep 21 2022

cp's OEIS Frontend

A227378 Smallest number with n = sum of distinct digits in decimal representation, cf. A217928.

Views

Author

Keywords

Comments

Programs

Haskell

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Python

Python

Scala

Smalltalk

Swift

Formula

Extensions

A216407 Sum of decimal digits not appearing in n.

Views

Author

Keywords

Links

Programs

Haskell

Mathematica

PARI

Formula

A280911 Numbers n such that sum of decimal digits of n equals number of prime divisors of n counted with multiplicity and sum of distinct decimal digits of n equals number of distinct primes dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A356981 Numbers k such that the sum of distinct digits of k equals the sum of the prime divisors of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

A357263 Numbers k such that the sum of the distinct digits of k is equal to the product of the prime divisors of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Extensions