A031298 -id:A031298 - cp's OEIS frontend

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

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

A030308 Triangle T(n, k): Write n in base 2, reverse order of digits, to get the n-th row.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1

Offset: 0

Clark Kimberling

This is the quite common, so-called "bittest" function, see PARI code. - M. F. Hasler, Jul 21 2013
For a given number m and a digit position k the corresponding sequence index n can be calculated by n(m, k) = m*(1 + floor(log_2(m))) - 2^(1 + floor(log_2(m))) + k + 1. For example: counted from right to left, the second digit of m = 13 (binary 1101) is '0'. Hence the sequence index is n = n(13, 2) = 39. - Hieronymus Fischer, May 05 2007
A070939(n) is the length of n-th row; A000120(n) is the sum of n-th row; A030101(n) is the n-th row seen as binary number; A000035(n) = T(n, 0). - Reinhard Zumkeller, Jun 17 2012

			Triangle begins :
0
1
0, 1
1, 1
0, 0, 1
1, 0, 1
0, 1, 1
1, 1, 1
0, 0, 0, 1
1, 0, 0, 1 - _Philippe Deléham_, Oct 12 2011

Reinhard Zumkeller, Rows n = 0..1023 of triangle, flattened
Index entries for sequences related to binary expansion of n

Cf. A030190.

Cf. A030341, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-3 to base-10 analogs.

a030308 n k = a030308_tabf !! n !! k
a030308_row n = a030308_tabf !! n
a030308_tabf = iterate bSucc [0] where
   bSucc []       = [1]
   bSucc (0 : bs) = 1 : bs
   bSucc (1 : bs) = 0 : bSucc bs
-- Reinhard Zumkeller, Jun 17 2012

A030308_row := n -> op(convert(n,base, 2)):
seq(A030308_row(n), n=0..23); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n, 2]], {n, 0, 23}]] (* T. D. Noe, Oct 12 2011 *)

A030308(n,k)=bittest(n,k) \\ Assuming that columns are numbered starting with k=0, as suggested by the formula from R. Zumkeller. - M. F. Hasler, Jul 21 2013

for n in range(20): print([int(z) for z in str(bin(n)[2:])[::-1]]) # Indranil Ghosh, Mar 31 2017

A030308_row = lambda n: n.bits() if n > 0 else [0]
for n in (0..23): print(A030308_row(n)) # Peter Luschny, Nov 28 2017

(0 to 31).map(Integer.toString(, 2).reverse).mkString.split("").map(Integer.parseInt()).toList // Alonso del Arte, Feb 10 2020

a(n) = floor(m/2^(k - 1)) mod 2, where m = max(j|A001855(j) < n) and k = n - A001855(m). - Hieronymus Fischer, May 05 2007, Sep 10 2007

T(n, k) = (n // 2^k) mod 2, for 0 <= k <= log[2](n) and n > 0; T(0, 0) = 0. ('//' denotes integer division). - Peter Luschny, Apr 20 2023

Initial 0 and better name by Philippe Deléham, Oct 12 2011

A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

			Triangle begins :
0
1
2
0, 1
1, 1
2, 1
0, 2
1, 2
2, 2
0, 0, 1
1, 0, 1
2, 0, 1
0, 1, 1
1, 1, 1
2, 1, 1 ...

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A081604 (row lengths), A053735 (row sums), A007089, A003137.

Cf. A030308, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

a030341 n k = a030341_tabf !! n !! k
a030341_row n = a030341_tabf !! n
a030341_tabf = iterate succ [0] where
   succ []     = [1]
   succ (2:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Feb 21 2013

A030341_row := n -> op(convert(n, base, 3)):
seq(A030341_row(n), n=0..32); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n,3]],{n,0,40}]] (* Harvey P. Dale, Oct 20 2014 *)

A030341(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\3^k%3 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030567 and others. - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset: 0

J. H. Conway, Dec 03 2007

We take 0^0 = 1.
The fixed points are in A135385.
For 1-digit or 2-digit numbers this is the same as A075877. - R. J. Mathar, Mar 28 2012
a(A221221(n)) = A133048(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

			20 -> 2^0 = 1,
21 -> 2^1 = 2,
24 -> 2^4 = 16,
39 -> 3^9 = 19683,
623 -> 6^2*3 = 108,
etc.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A075877, A133501 (number of steps to reach fixed point), A133502, A135385 (the conjectured list of fixed points), A135384 (numbers which converge to 2592). For records see A133504, A133505; for the fixed points that are reached when this map is iterated starting at n, see A287877.
Cf. also A133048 (powerback), A031346 and A003001 (persistence).
Cf. also A031298, A007376.

a133500 = train . reverse . a031298_row where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

powertrain:=proc(n) local a,i,n1,n2,t1,t2; n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=1; for i from 0 to floor(t2/2)-1 do a := a*t1[t2-2*i]^t1[t2-2*i-1]; od: if t2 mod 2 = 1 then a:=a*t1[1]; fi; RETURN(n2*a); end; # N. J. A. Sloane, Dec 03 2007

ptm[n_]:=Module[{idn=IntegerDigits[n]},If[EvenQ[Length[idn]],Times@@( #[[1]]^ #[[2]] &/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]] &/@ Partition[ Most[idn],2]))Last[idn]]]; Array[ptm,70,0] (* Harvey P. Dale, Jul 15 2019 *)

def A133500(n):
    s = str(n)
    l = len(s)
    m = int(s[-1]) if l % 2 else 1
    for i in range(0,l-1,2):
        m *= int(s[i])**int(s[i+1])
    return m # Chai Wah Wu, Jun 16 2017

A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 0, 1, 1, 1, 1, 1, 2

Offset: 0

Clark Kimberling

If columns are numbered starting with k=0, then T(n,k) contains the coefficient of 6^k in n's base-6 expansion. - M. F. Hasler, Jul 21 2013

R. J. Mathar, Table of n, a(n) for n = 0..5950

See A030548 for a quite complete list of crossreferences.
Cf. A030568 - A030573 for positions of a given digit.
Cf. A030575 - A030580 for run lengths, A030581 - A030585 for more.
Row sums (same as those of A030548) are in A053827.
Cf. A030308, A030341, A030386, A031235, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,6]],{n,0,50}]] (* Harvey P. Dale, Sep 27 2015 *)

A030567(n,k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\6^k%6 \\ Assuming that columns start with k=0, cf. comment. TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

Initial 0 and better name from Philippe Deléham, Oct 20 2011

Edited and crossrefs added by M. F. Hasler, Jul 21 2013

A031235 Triangle T(n,k): write n in base 5, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A030386, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Cf. A007091.

a031235 n k = a031235_tabf !! n !! k
a031235_row n = a031235_tabf !! n
a031235_tabf = iterate succ [0] where
   succ []     = [1]
   succ (4:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Sep 18 2015

Reverse[IntegerDigits[#,5]]&/@Range[0,40]//Flatten (* Harvey P. Dale, Aug 02 2016 *)

A031235(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\5^k%5 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A030386 Triangle T(n,k): write n in base 4, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

			Triangle begins:
0
1
2
3
0, 1
1, 1
2, 1
3, 1
0, 2
1, 2
2, 2
3, 2
0, 3
1, 3
2, 3
3, 3
0, 0, 1
1, 0, 1 ... - _Philippe Deléham_, Oct 20 2011

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Cf. A007090.

a030386 n k = a030386_tabf !! n !! k
a030386_row n = a030386_tabf !! n
a030386_tabf = iterate succ [0] where
   succ []     = [1]
   succ (3:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Sep 18 2015

A030386_row := n -> op(convert(n, base, 4)):
seq(A030386_row(n), n=0..36); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n,4]],{n,0,50}]] (* Harvey P. Dale, Oct 13 2012 *)

A030386(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\4^k%4 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... \\ M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

cp's OEIS Frontend

A031309 a(n)=least k such that base 10 representation of n begins at s(k), where s=A031298.

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

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A030308 Triangle T(n, k): Write n in base 2, reverse order of digits, to get the n-th row.

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A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

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A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^b*c^d* ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

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A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

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A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.