A051885 -id:A051885 - cp's OEIS frontend

A060712 Smallest number whose sum of digits is 3^n.

1, 3, 9, 999, 999999999, 999999999999999999999999999, 999999999999999999999999999999999999999999999999999999999999999999999999999999999

Offset: 0

Robert G. Wilson v, Apr 21 2001

Harry J. Smith, Table of n, a(n) for n = 0..8

Cf. A051885.

Do[ a = {}; While[ Apply[ Plus, a ] + 9 < 3^n, a = Append[ a, 9 ] ]; If[ Apply[ Plus, a ] != 3^n, a = Prepend[ a, 3^n - Apply[ Plus, a ] ] ]; Print[ FromDigits[ a ] ], {n, 1, 6} ]
Join[{1,3},Table[FromDigits[PadRight[{},3^(n-2),9]],{n,2,6}]] (* Harvey P. Dale, Jun 10 2015 *)

a(n)={ my(s=3^n, x=s\9, d=s-9*x); (d+1)*10^x - 1 } \\ Harry J. Smith, Jul 10 2009

a(n) = A051885(3^n). - Andrew Howroyd, Dec 08 2024

a(n) = 10^(3^(n-2)) - 1 for n > 1. - Stefano Spezia, Mar 27 2025

A077495 a(n) = smallest k such that the digit sum of 8k is n.

Original entry on oeis.org

0, 125, 25, 15, 5, 4, 3, 2, 1, 9, 8, 7, 6, 23, 22, 12, 11, 37, 36, 62, 61, 87, 86, 112, 111, 236, 361, 486, 611, 736, 861, 986, 1111, 1236, 2486, 3736, 4986, 6236, 7486, 8736, 9986, 11236, 12486, 24986, 37486, 49986, 62486, 74986, 87486, 99986, 112486, 124986

Offset: 0

Amarnath Murthy, Nov 07 2002

A069536(n)/8.

Cf. A051885, A069532, A069534, A069536, A077493, A077494.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a077495 n = fromJust $ elemIndex n $ map a007953 a008590_list
a077495_list = map a077495 [0..]
-- Reinhard Zumkeller, Dec 09 2011

From Robert Israel, Nov 19 2022: (Start) G.f.: -x^24*(985*x^9 - 125*x^8 - 125*x^7 - 125*x^6 - 125*x^5 - 125*x^4 - 125*x^3 - 125*x^2 - 125*x - 111)/((x - 1)*(10*x^9 - 1)) + 112*x^23 + 86*x^22 + 87*x^21 + 61*x^20 + 62*x^19 + 36*x^18 + 37*x^17 + 11*x^16 + 12*x^15 + 22*x^14 + 23*x^13 + 6*x^12 + 7*x^11 + 8*x^10 + 9*x^9 + x^8 + 2*x^7 + 3*x^6 + 4*x^5 + 5*x^4 + 15*x^3 + 25*x^2 + 125*x.

For n >= 24, a(n) = 125*A051885(n-24) + 111. (End)

Corrected and extended by Ray Chandler, Aug 03 2003

Missing a(0)=0 added and offset adjusted by Reinhard Zumkeller, Dec 09 2011

A061219 a(n) is the largest number which can be formed with no zeros, using least number of digits and having digit sum = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 91, 92, 93, 94, 95, 96, 97, 98, 99, 991, 992, 993, 994, 995, 996, 997, 998, 999, 9991, 9992, 9993, 9994, 9995, 9996, 9997, 9998, 9999, 99991, 99992, 99993, 99994, 99995, 99996, 99997, 99998, 99999, 999991, 999992, 999993, 999994

Offset: 1

Amarnath Murthy, Apr 22 2001

a(n) is the digit reversal of terms of A051885 giving such smallest numbers.

			a(22) = 994, digit sum = 22.
a(100) = 999999999991.

Michael S. Branicky, Table of n, a(n) for n = 1..9000

Cf. A051885.

dsn[n_]:=Module[{d=Quotient[n,9]},FromDigits[PadLeft[{n-9d},d,9]]]; If[Divisible[#,10],#/10,#]&/@Array[dsn,50,10] (* Harvey P. Dale, Dec 08 2013 *)

def a(n): return int("9"*(n//9)+str(n%9)*(n%9>0))
print([a(n) for n in range(1, 50)]) # Michael S. Branicky, Aug 16 2023

More terms from Harvey P. Dale, Dec 08 2013

Offset corrected by Michael S. Branicky, Aug 16 2023

A131668 Smallest number whose sum of digits is 2n+1.

Original entry on oeis.org

1, 3, 5, 7, 9, 29, 49, 69, 89, 199, 399, 599, 799, 999, 2999, 4999, 6999, 8999, 19999, 39999, 59999, 79999, 99999, 299999, 499999, 699999, 899999, 1999999, 3999999, 5999999, 7999999, 9999999, 29999999, 49999999, 69999999, 89999999, 199999999

Offset: 0

Paul Curtz, Oct 03 2007

Numbers which can't be represented as the sum of two numbers with the same sum of digits in base 10 (according to Daniel Starodubtsev). More generally, this definition and the definition from the name of this sequence matches for any even base. - Mikhail Kurkov, May 19 2019 [verification needed]

			For n=0, the least number with sum of digits 2*0+1=1 is 1, so a(0)=1.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,100,-100).

Cf. A051885, A133296.

a(n) = {my(k=0); while (sumdigits(k) != 2*n+1, k++); k;} \\ Michel Marcus, May 19 2019

a(n) = if(n<5, return(2*n+1)); n-=5;[30, 50, 70, 90, 200, 400, 600, 800, 1000][n%9+1] * 100^(n\9)-1 \\ David A. Corneth, May 19 2019

a(n) = h(n,10)*10^g(n,10)-1, with f(n,k) = floor((n+1)/(k-1)) - floor(n/(k-1)), g(n,k) = floor(2*(n+1)/(k-1)) - f(n,k), h(n,k) = 2*(n+1) - (k-1)*g(n,k). - Mikhail Kurkov, May 19 2019

A213653 Least semiprime whose digital sum is n, or 0 if no such integer exists.

Original entry on oeis.org

0, 10, 10001, 21, 4, 14, 6, 25, 26, 9, 46, 38, 39, 49, 77, 69, 169, 278, 0, 289, 299, 489, 589, 689, 699, 799, 899, 0, 2899, 3899, 4989, 5899, 5999, 6999, 7999, 9899, 0, 19999, 29999, 48999, 58999, 68999, 69999, 88999, 99899, 0, 299899, 398999, 589989

Offset: 0

Robert G. Wilson v, Jun 17 2012

a(9k) = 0 for all k>1.
I conjecture that all terms > 278, except for 10001, end in the digit "9". What is the next term a(n) > 69 violating monotony, i.e., such that a(n) < a(n-1)? M. F. Hasler, Jun 17 2012
a(88) = 7999999999 < a(87) = 8899899999. - Alois P. Heinz, Jun 17 2012
a(76) = 499999999 < a(75) = 597999999. - Donovan Johnson, Jun 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..400 (first 101 terms from Donovan Johnson)

semiPrimeQ[n_] := PrimeOmega[n] == 2; t = Table[0, {100}]; k = 1; While[k < 10^7, If[ semiPrimeQ@ k, s = Plus @@ IntegerDigits@ k; If[s < 101 && t[[s]] == 0, t[[s]] = k; Print[{s, k}]]]; k++]

A213653(n)={ n%9 || n==9 || return; forstep( a=A051885(n),9e9,9, bigomega(a)==2 || next; A007953(a)==n & return(a))} \\ - M. F. Hasler, Jun 17 2012

a(0)-a(62) double-checked with given PARI code by M. F. Hasler, Jun 17 2012

A259046 Smallest m such that A259043(m) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 15, 25, 16, 26, 17, 27, 18, 59, 78, 69, 88, 79, 98, 89, 108, 99, 509, 618, 609, 718, 709, 818, 809, 918, 909, 5009, 6018, 6009, 7018, 7009, 8018, 8009, 9018, 9009, 50009, 60018, 60009, 70018, 70009, 80018, 80009, 90018

Offset: 0

Reinhard Zumkeller, Jun 17 2015

nonn

A259043(a(n)) = n and A259043(m) != n for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..76

Cf. A259043, A051885.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a259046 = fromJust . (`elemIndex` (map a259043 [0..]))

f(n) = if (n<10, n, my(u=n%10); f(n\10 + u) + u); \\ A259043
a(n) = my(m=0); while (f(m)!=n, m++); m; \\ Michel Marcus, Jan 23 2022

A268605 a(1) = 0; a(n+1) is the smallest integer in which the difference between its digits sum and the a(n) digits sum is equal to the n-th prime.

Original entry on oeis.org

0, 2, 5, 19, 89, 1999, 59999, 4999999, 599999999, 199999999999, 399999999999999, 799999999999999999, 8999999999999999999999, 499999999999999999999999999, 29999999999999999999999999999999, 4999999999999999999999999999999999999

Offset: 1

Francesco Di Matteo, Feb 17 2016

First 8 terms are primes (and are also in A061248). Next terms are not always primes.

			a(4) = 19 and 1 + 9 = 10; so a(5) = 89 because 8 + 9 = 17 and 17 - 10 = 7, that is the 4th prime.

Francesco Di Matteo, Table of n, a(n) for n = 1..25

Cf. A269306, A061248, A067180, A051885.

findnext(x, k) = {sx = sumdigits(x); pk = prime(k); y = 1; while (sumdigits(y) - sx != pk, y++); y;}
lista(nn) = {print1(x = 0, ", "); for (k=1, nn, y = findnext(x, k); print1(y, ", "); x = y;);} \\ Michel Marcus, Feb 19 2016

sumprime = 0
isPrime=lambda x: all(x % i != 0 for i in range(int(x**0.5)+1)[2:])
print(0)
for i in range(2,100):
  if isPrime(i):
    alfa = ""
    k = i + sumprime
    sumprime = k
    while k > 9:
      alfa = alfa + "9"
      k = k - 9
    alfa = str(k)+alfa
    print(alfa)

a(n) = A051885( A007504(n-1) ). - R. J. Mathar, Jun 19 2021

NAME adapted to offset by R. J. Mathar, Jun 19 2021

A307560 a(n) = smallest m such that A307629(m) = n.

Original entry on oeis.org

0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 29, 39, 49, 59, 69, 79, 89, 99, 10000000000000000000, 109, 1006, 119, 100000000000000000000000, 129, 100004, 139, 1008, 149, 100000000000000000000000000000, 159, 10000000000000000000000000000000, 169, 1019, 179, 100006

Offset: 0

Arran Ireland, Apr 14 2019

			a(39) = 1039 as (1 + 0) + (1 + 3) + (1 + 9) + (0 + 3) + (0 + 9) + (3 + 9) = 39. The sums in brackets are pairs of digits of 1039. No positive integer less than 1039 has this pairwise digit sum. - _David A. Corneth_, Apr 16 2019

Arran Ireland, Table of n, a(n) for n = 0..304 (a(0) inserted by _Georg Fischer_, May 05 2019)
Arran Ireland, Python program (a(0) inserted by _Georg Fischer_, May 05 2019)
Reddit website, Integer sequence: Pairwise Combinatorial Digit Sum

Cf. A051885, A179239, A307629.

for n in [1..50] do for d in Divisors(n) do if n le 9*d*(d+1) then nd:=d+1; sdLeft:=n div d; S:=[]; for j in [1..nd-1] do if sdLeft gt 9 then S[j]:=9; else S[j]:=sdLeft-1; end if; sdLeft-:=S[j]; end for; S[nd]:=sdLeft; a:=Seqint(S); n, a; break; end if; end for; end for; // Jon E. Schoenfield, Apr 15 2019

fs[nd_, s_] := If[nd*9 < s, 0, Block[{n=10^(nd-1), f=0}, While[n < 10^nd, If[Total@ IntegerDigits@ n == s, f = n; Break[], n++]]; f]]; a[n_] := Block[{s}, Do[s = fs[d+1, n/d]; If[s > 0, Break[]], {d, Divisors[n]}]; s];  Join[{0}, Array[a,50]] (* Giovanni Resta, Apr 15 2019 *)

Let d be the smallest divisor of n for which 9*d*(d+1) >= n; then a(n) is the smallest (d+1)-digit number whose digit sum is n/d. - Jon E. Schoenfield, Apr 15 2019

A342810 Numbers k that divide the smallest number whose sum of digits is k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 27, 81, 191, 243, 729, 999, 2187, 2997, 6561, 8991, 19683, 26973, 33321, 36963, 39049, 59049, 80919, 100389, 110889, 118827, 177147, 177897, 183951, 242757, 332667, 356481, 531441, 551853, 728271, 998001, 1069443, 1367631, 1594323, 1655559, 2184813

Offset: 1

Ruediger Jehn, Mar 22 2021

By definition, if k divides A051885(k), then k is a term of this sequence.
From Ruediger Jehn, Jun 17 2021: (Start)
None of the terms is divisible by 2*5*11*13.
If a term x has the form 3^m * y where m > 1 (which is the case for the overwhelming number of terms of this sequence), then all prime factors of y are terms of A066364.
If a term x has the form 3^m * p * q where m > 1, where p is a term of A066364 and where q is the product of all other factors of the prime factorization of x, then all numbers 3^m * p^i * q are also terms for any integer i. (End)

			21 is a term because the smallest number with a digital sum of 21 is 399 (A051885(21) = 399) which is divisible by 21.

Kester Habermann, Table of n, a(n) for n = 1..792 (first 200 terms from Chai Wah Wu)
Rüdiger Jehn and Kester Habermann, Properties of terms of OEIS A342810, arXiv:2106.05866 [math.GM], 2021.

Cf. A007953, A051885, A066364.

MAX=10000; for (e = 0, MAX, for (d = 1, 9, k =(d+1)*10^e - 1; x = d+9*e; if (k%x==0, print1(x, ", ");)))

A342810_list = [n for n in range(1,10**6) if n==1 or ((n % 9)+1)*pow(10,n//9,n) % n == 1] # Chai Wah Wu, Apr 04 2021

Name clarified by Jon E. Schoenfield, Apr 27 2021

A342829 a(n) is the smallest number whose sum of digits is equal to the sum of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 8

Offset: 1

Michel Marcus, Mar 23 2021

			For n=10, A007953(10) = 1, and 1 is the smallest integer x satisfying A007953(x)=1, so a(10)=1.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A007953 (sum of digits), A051885.

a(n) = my(k=1, s=sumdigits(n)); while(sumdigits(k) != s, k++); k;

a(n) = A051885(A007953(n)).

cp's OEIS Frontend

A060712 Smallest number whose sum of digits is 3^n.

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A077495 a(n) = smallest k such that the digit sum of 8k is n.

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A061219 a(n) is the largest number which can be formed with no zeros, using least number of digits and having digit sum = n.

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A131668 Smallest number whose sum of digits is 2n+1.

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A213653 Least semiprime whose digital sum is n, or 0 if no such integer exists.

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A259046 Smallest m such that A259043(m) = n.

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A268605 a(1) = 0; a(n+1) is the smallest integer in which the difference between its digits sum and the a(n) digits sum is equal to the n-th prime.

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A307560 a(n) = smallest m such that A307629(m) = n.

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