A053735 -id:A053735 - cp's OEIS frontend

A230856 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly four solutions.

248, 492, 978, 1222, 1708, 1952, 2192, 2196, 2436, 2680, 3166, 3410, 3896, 4140, 4380, 4384, 4624, 4868, 5354, 5598, 6084, 6328, 6566, 6572, 6810, 7054, 7540, 7784, 8270, 8514, 8754, 8758, 8998, 9242, 9728, 9972, 10458, 10702, 10942, 10946, 11186, 11430, 11916, 12160, 12646, 12890, 13128

Offset: 1

N. J. A. Sloane, Oct 31 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 2, 3, 10, 5, 2, 5, 7, 3, 21, 5, 10, 36, 14, 5, 27, 12, 2, 5, 16, 5, 14, 23, 7, 12, 29, 3, 21, 5, 21, 78, 27, 5, 27, 12, 10, 78, 14, 36, 136, 44, 14, 90, 25, 5, 27, 23, 27, 90, 61, 12, 42, 38, 2, 5, 16, 5, 14, 23, 16, 23, 67, 5, 27, 23, 14, 44, 40, 23, 61, 80, 7, 12, 67, 12, 25, 80, 29, 38, 121, 3, 21, 5, 21, 78, 27, 5, 27, 12, 21, 465, 27, 78, 300, 90, 27

Offset: 0

Antti Karttunen, Jul 25 2017

nonn

For all i, j: a(i) = a(j) => A006047(i) = A006047(j) => A053735(i) = A053735(j).

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A000027, A278222, A289813, A289814, A290091, A290092.

Cf. A006047, A053735, A290079 (some of the matched sequences).

(define (A290093 n) (* (/ 1 2) (+ (expt (+ (A290091 n) (A290092 n)) 2) (- (A290091 n)) (- (* 3 (A290092 n))) 2)))

a(n) = (1/2)*(2 + ((A290091(n)+A290092(n))^2) - A290091(n) - 3*A290092(n)).

A033095 Number of 1's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

1, 1, 3, 4, 6, 6, 9, 6, 10, 10, 12, 11, 16, 13, 15, 14, 16, 13, 18, 15, 21, 20, 21, 16, 24, 20, 23, 23, 26, 25, 32, 22, 26, 25, 25, 28, 34, 28, 32, 30, 35, 30, 37, 31, 35, 36, 35, 31, 41, 34, 37, 36, 39, 35, 43, 38, 44, 41, 42, 38, 49, 40, 43

Offset: 1

Clark Kimberling

Cf. A053735, A053737, A062756, A077267.

Cf. A033093, A033096, A033097, A033099, A033101, A033103, A033105, A033107, A033109, A033111.

f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 1]; Array[f, 63] (* Robert G. Wilson v, Nov 14 2012 *)

G.f.: x+(Sum_{b>=2} (Sum_{k>=0} x^(b^k)/(Sum_{0<=iFranklin T. Adams-Watters, Nov 03 2005

A053831 Sum of digits of n written in base 11.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10}, 1->{1,2,3,4,5,6,7,8,9,10,11}, 2->{2,3,4,5,6,7,8,9,10,11,12}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1 + 9 = 10 because 20 is written as 19 base 11.

Tanar Ulric, Table of n, a(n) for n = 0..10000
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.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A007953, A064458, A138530.

Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, this sequence, A053832, A053833, A053834, A053835, A053836.

int Base11DigitSum(int n) {
   int count = 0;
   while (n != 0) { count += n % 11; n = n / 11; }
   return count;
} // Tanar Ulric, Oct 20 2021

Table[Plus @@ IntegerDigits[n, 11], {n, 0, 86}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 10}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

a(n)=if(n<1,0,if(n%11,a(n-1)+1,a(n/11)))

a(n)=sumdigits(n,11) \\ Charles R Greathouse IV, Oct 20 2021

From Benoit Cloitre, Dec 19 2002: (Start)
a(0)=0, a(11n+i) = a(n)+i for 0 <= i <= 10.
a(n) = n-(m-1)*(Sum_{k>0} floor(n/m^k)) = n-(m-1)*A064458(n). (End)
a(n) = A138530(n,11) for n > 10. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 11*log(11)/10 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A173525 a(n) = 1 + A053824(n-1), where A053824 = sum of digits in base 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12

Offset: 1

Omar E. Pol, Feb 20 2010

Also: a(n) = A053824(5^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053824. (See the comment by M. F. Hasler for the proof.)
This means: if A053824 is regarded as a triangle then the rows converge to this sequence.
See conjecture in the entry A000120, and the case of base 2 in A063787.
From R. J. Mathar, Dec 09 2010: (Start)
In base b=5, A053824 starts counting up from 1 each time the index wraps around a power of b: A053824(b^k)=1.
Obvious recurrences are A053824(m*b^k+i) = m+A053824(i), 1 <= m < b-1, 0 <= i < b^(k-1).
So A053824 can be decomposed into a triangle T(k,n) = A053824(b^k+n-1), assuming that column indices start at n=1; row lengths are (b-1)*b^k.
There is a self-similarity in these sequences; a sawtooth structure of periodicity b is added algebraically on top of a sawtooth structure of periodicity b^2, on top of a periodicity b^3 etc. This leads to some "fake" finitely periodic substructures in the early parts of each row of T(.,.): often, but not always, a(n+b)=1+a(n). Often, but not always, a(n+b^2)=1+a(n) etc.
The common part of the rows T(.,.) grows with the power of b as shown in the recurrence above, and defines a(n) in the limit of large row indices k. (End)
The two definitions agree because the first 5^r terms in each row correspond to numbers 5^r, 5^r+1,...,5^r+(5^r-1), which are written in base 5 as a leading 1 plus the digits of 0,...,5^r-1. - M. F. Hasler, Dec 09 2010
From Omar E. Pol, Dec 10 2010: (Start)
In the scatter plots of these sequences, the basic structure is an element with b^2 points, where b is the associated base. (Scatter plots are created with the "graph" button of a sequence.) Sketches of these structures look as follows, the horizontal axis a squeezed version of the index n, b consecutive points packed vertically, and the vertical axis a(n):
........................................................
................................................ * .....
............................................... ** .....
..................................... * ...... *** .....
.................................... ** ..... **** .....
.......................... * ...... *** .... ***** .....
......................... ** ..... **** ... ****** .....
............... * ...... *** .... ***** ... ***** ......
.............. ** ..... **** .... **** .... **** .......
.... * ...... *** ..... *** ..... *** ..... *** ........
... ** ...... ** ...... ** ...... ** ...... ** .........
... * ....... * ....... * ....... * ....... * ..........
........................................................
... b=2 ..... b=3 ..... b=4 ..... b=5 ..... b=6 ........
. A000120 . A053735 . A053737 . A053824 . A053827 ......
. A063787 . A173523 . A173524 . A173525 . A173526 ......
........................................................
............................................. * ........
............................................ ** ........
........................... * ............. *** ........
.......................... ** ............ **** ........
........... *............ *** ........... ***** ........
.......... ** .......... **** .......... ****** ........
......... ***.......... ***** ......... ******* ........
........ **** ........ ****** ........ ******** ........
....... ***** ....... ******* ....... ********* ........
...... ****** ...... ******** ....... ******** .........
..... ******* ...... ******* ........ ******* ..........
..... ****** ....... ****** ......... ****** ...........
..... ***** ........ ***** .......... ***** ............
..... **** ......... **** ........... **** .............
..... *** .......... *** ............ *** ..............
..... ** ........... ** ............. ** ...............
..... * ............ * .............. * ................
........................................................
..... b=7 .......... b=8 ............ b=9 ..............
... A053828 ...... A053829 ........ A053830 ............
... A173527 ...... A173528 ........ A173529 ............(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..3126=5^5+1
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A000120, A053824, A063787, A173523, A173524, A173526, A173527, A173528, A173529.

a173525 = (+ 1) . a053824 . (subtract 1) -- Reinhard Zumkeller, Jan 31 2014

A053825 := proc(n) add(d, d=convert(n,base,5)) ; end proc:
A173525 := proc(n) local b,k; b := 5 ; if n < b then n; else k := n/(b-1);   k := ceil(log(k)/log(b)) ; A053825(b^k+n-1) ; end if; end proc:
seq(A173525(n),n=1..100) ;

Total[IntegerDigits[#,5]]+1&/@Range[0,100] (* Harvey P. Dale, Jun 14 2015 *)

A173525(n)={ my(s=1); n--; until(!n\=5, s+=n%5); s } \\ M. F. Hasler, Dec 09 2010

A173525(n)={ my(s=1+(n=divrem(n-1,5))[2]); while((n=divrem(n[1],5))[1],s+=n[2]); s+n[2] } \\ M. F. Hasler, Dec 09 2010

a(n) = A053824(5^k + n - 1) where k >= ceiling(log_5(n/4)). - R. J. Mathar, Dec 09 2010

More terms from Vincenzo Librandi, Aug 02 2010

A240236 Triangle read by rows: sum of digits of n in base k, for 2<=k<=n.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 3, 2, 1, 3, 3, 4, 3, 2, 1, 1, 4, 2, 4, 3, 2, 1, 2, 1, 3, 5, 4, 3, 2, 1, 2, 2, 4, 2, 5, 4, 3, 2, 1, 3, 3, 5, 3, 6, 5, 4, 3, 2, 1, 2, 2, 3, 4, 2, 6, 5, 4, 3, 2, 1, 3, 3, 4, 5, 3, 7, 6, 5, 4, 3, 2, 1, 3, 4, 5, 6, 4, 2, 7, 6, 5, 4, 3, 2, 1

Offset: 2

Franklin T. Adams-Watters, Apr 02 2014

			Triangle starts:
  1
  2 1
  1 2 1
  2 3 2 1
  2 2 3 2 1
  3 3 4 3 2 1

Reinhard Zumkeller, Rows n = 2..100 of triangle, flattened

Row sums give A043306.
Cf. A000120, A053735, A053737, A007953.
See A138530 for another version.
Cf. A002260, A090623.

a240236 n k = a240236_tabl !! (n-1) !! (k-1)
a240236_row n = a240236_tabl !! (n-1)
a240236_tabl = zipWith (map . flip q)
                       [2..] (map tail $ tail a002260_tabl) where
   q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
-- Reinhard Zumkeller, Apr 29 2015

Table[Total[Flatten[IntegerDigits[n,k]]],{n,20},{k,2,n}]//Flatten (* Harvey P. Dale, Jan 13 2025 *)

T(n,k) = local(r=0);if(k<2,-1,while(n>0,r+=n%k;n\=k);r)

T(n, k) = sumdigits(n, k) \\ Zhuorui He, Aug 25 2025

T(n,k) = n - (k - 1) * Sum_{i=1..floor(log_k(n))} floor(n/k^i). - Ridouane Oudra, Sep 27 2024

T(n,k) = n - (k - 1) * A090623(n,k). - Zhuorui He, Aug 25 2025

A290094 Restricted growth sequence transform of A290093.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 3, 5, 6, 2, 7, 5, 4, 8, 9, 5, 10, 11, 3, 5, 12, 5, 9, 13, 6, 11, 14, 2, 7, 5, 7, 15, 10, 5, 10, 11, 4, 15, 9, 8, 16, 17, 9, 18, 19, 5, 10, 13, 10, 18, 20, 11, 21, 22, 3, 5, 12, 5, 9, 13, 12, 13, 23, 5, 10, 13, 9, 17, 24, 13, 20, 25, 6, 11, 23, 11, 19, 25, 14, 22, 26, 2, 7, 5, 7, 15, 10, 5, 10, 11, 7, 27, 10, 15, 28, 18, 10, 29, 21, 5, 10, 13

Offset: 0

Antti Karttunen, Jul 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..59049

For all i, j: a(i) = a(j) <=> A290093(n) = A290093(n), thus this matches to all the same base-3 (ternary) related sequences as A290093: A006047, A053735, A062756, A081603, A117942, A206424, A227428, A290091, A290092, A290079, and many others.

A000989 3-adic valuation of binomial(2n, n): largest k such that 3^k divides binomial(2n, n).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 0, 3, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 2, 3, 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 0, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 3, 3, 4, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 3, 3

Offset: 0

N. J. A. Sloane, R. K. Guy

a(n) = 0 if and only if n is in A005836. - Charles R Greathouse IV, May 19 2013
sign(a(n+1) - a(n)) is repeat [0, 1, -1]. - Filip Zaludek, Oct 29 2016
By Kummer's theorem, number of carries when adding n + n in base 3. - Robert Israel, Oct 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Michael Gilleland, Some Self-Similar Integer Sequences.
E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, Vol. 44 (1852), pp. 93-146; alternative link.
Dorel Miheţ, Legendre's and Kummer's theorems again, Resonance, Vol. 15, No. 12 (2010), pp. 1111-1121; alternative link.
Armin Straub, Victor H. Moll and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arithmetica, Vol. 140, No. 1 (2009), pp. 31-42.
Wikipedia, Kummer's theorem.

Cf. A000984, A005836, A007949, A053735, A067397.

a000989 = a007949 . a000984  -- Reinhard Zumkeller, Nov 19 2015

f:= proc(n) option remember; local k,m,d;
   k:= floor(log[3](n));
   d:= floor(n/3^k);
   m:= n-d*3^k;
   if d = 2 or 2*m > 3^k then procname(m)+1
   else procname(m)
   fi
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Oct 30 2016

p=3; Array[ If[ Mod[ bi=Binomial[ 2#, # ], p ]==0, Select[ FactorInteger[ bi ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 27*3, 0 ]
Table[ IntegerExponent[ Binomial[2 n, n], 3], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2016 *)

PARI
```
a(n) = valuation(binomial(2*n, n), 3)
```

a(n)=my(N=2*n,s);while(N\=3,s+=N);while(n\=3,s-=2*n);s \\ Charles R Greathouse IV, May 19 2013

a(n) = Sum_{k>=0} floor(2*n/3^k) - 2*Sum_{k>=0} floor(n/3^k). - Benoit Cloitre, Aug 26 2003
a(n) = A007949(A000984(n)). - Reinhard Zumkeller, Nov 19 2015
From Robert Israel, Oct 30 2016: (Start)
If 2*n < 3^k then a(3^k+n) = a(n).
If n < 3^k < 2*n then a(3^k+n) = a(n)+1.
If n < 3^k then a(2*3^k+n) = a(n)+1. (End)
a(n) = A053735(n) - A053735(2*n)/2. - Amiram Eldar, Feb 12 2021
From Miles Wilson, Jul 06 2025: (Start)
a(n) = A007949(n+1) + A067397(n).
G.f.: Sum_{k>=1} (x^(3^k/2+1/2)-x^(3^k))/((x-1)*(x^(3^k)-1)). (End)

A006287 Sum of squares of digits of ternary representation of n.

Original entry on oeis.org

0, 1, 4, 1, 2, 5, 4, 5, 8, 1, 2, 5, 2, 3, 6, 5, 6, 9, 4, 5, 8, 5, 6, 9, 8, 9, 12, 1, 2, 5, 2, 3, 6, 5, 6, 9, 2, 3, 6, 3, 4, 7, 6, 7, 10, 5, 6, 9, 6, 7, 10, 9, 10, 13, 4, 5, 8, 5, 6, 9, 8, 9, 12, 5, 6, 9, 6, 7

Offset: 0

N. J. A. Sloane

Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 0..500
Michael Gilleland, Some Self-Similar Integer Sequences
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A053735

A006287 := proc(n)
    local d ;
    convert(n,base,3) ;
    add(d^2,d=%) ;
end proc: # R. J. Mathar, Jun 13 2014

Table[Total[IntegerDigits[n,3]^2],{n,0,80}]  (* Harvey P. Dale, Apr 23 2011 *)

A239619 Base 3 sum of digits of prime(n).

Original entry on oeis.org

2, 1, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 5, 5, 5, 7, 5, 5, 5, 7, 5, 7, 3, 5, 5, 5, 5, 7, 3, 5, 5, 7, 5, 5, 7, 7, 7, 3, 5, 5, 7, 5, 5, 5, 7, 5, 7, 7, 7, 7, 9, 9, 9, 5, 5, 5, 7, 3, 5, 5, 5, 7, 5, 7, 7, 7, 5, 5, 7, 7, 5, 7, 7, 7, 5, 7, 7, 7, 9, 5, 7, 7, 9, 5, 7, 7, 9

Offset: 1

Tom Edgar, Mar 22 2014

			The fifth prime is 11, 11 in base 3 is (1,0,2) so a(5)=1+0+2=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A007605, A014499, A053735.

[&+Intseq(NthPrime(n),3): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 3], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = vecsum(digits(prime(n), 3)); \\ Michel Marcus, Mar 07 2020

[sum(i.digits(base=3)) for i in primes_first_n(200)]

a(n) = A053735(A000040(n)).

cp's OEIS Frontend

A230856 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly four solutions.

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A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A033095 Number of 1's when n is written in base b for 2<=b<=n+1.

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A053831 Sum of digits of n written in base 11.

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A173525 a(n) = 1 + A053824(n-1), where A053824 = sum of digits in base 5.

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A240236 Triangle read by rows: sum of digits of n in base k, for 2<=k<=n.

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A290094 Restricted growth sequence transform of A290093.

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A000989 3-adic valuation of binomial(2*n, n): largest k such that 3^k divides binomial(2*n, n).

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A000989 3-adic valuation of binomial(2n, n): largest k such that 3^k divides binomial(2n, n).