A030567 -id:A030567 - cp's OEIS frontend

A030548 Write n in base 6 and juxtapose.

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

Offset: 1

Clark Kimberling

Base-6 analog of what in base 7 is A030998, in base 10 is A007376. In general, the Barbier infinite word base n (in this case, 6). - Jonathan Vos Post, May 13 2007
An irregular table in which the n-th row lists the base-6 digits of n. - Jason Kimberley, Dec 07 2012
The base-6 Champernowne constant: It is normal in base 6. - Jason Kimberley, Dec 07 2012

R. J. Mathar, Table of n, a(n) for n = 1..5950

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), this sequence (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A030567 for the same table with reversed rows.

&cat[Reverse(IntegerToSequence(n,6)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 50, 6] (* or *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ a[#, 6] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A030548_gen(): return chain.from_iterable(digits(m, 6)[1:] for m in count(1))
A030548_list = list(islice(A030548_gen(), 30)) # Chai Wah Wu, Jan 07 2022

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

The length of n-th row is given in A055642(n). - Reinhard Zumkeller, Jul 04 2012

According to the formula for T(n,1), columns are numbered starting with 1. One might also number columns starting with the offset 0, as to have the coefficient of 10^k in column k. - M. F. Hasler, Jul 21 2013

Reinhard Zumkeller, Rows n = 0..2500 of triangle, flattened

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031087 for the base-2 to base-9 analogs.

a031298 n k = a031298_tabf !! n !! k
a031298_row n = a031298_tabf !! n
a031298_tabf = iterate succ [0] where
   succ []     = [1]
   succ (9:ds) = 0 : succ ds
   succ (d:ds) = (d + 1) : ds
-- Reinhard Zumkeller, Jul 04 2012

Table[Reverse[IntegerDigits[n]],{n,0,50}]//Flatten (* Harvey P. Dale, Mar 07 2023 *)

T(n,k)=n\10^(k-1)%10 \\ M. F. Hasler, Jul 21 2013

T(n,1) = A010879(n); T(n,A055642(n)) = A000030(n). - Reinhard Zumkeller, Jul 04 2012

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

Edited 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

A053827 Sum of digits of (n written in base 6).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10

Offset: 0

Henry Bottomley, Mar 28 2000

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

Sum of six consecutive terms is (15,21,27,33,39,45; 21,27,33,39,45,51; 27,33,39,45,51,57; and so on). - Vincenzo Librandi, Aug 02 2010

			a(20)=3+2=5 because 20 is written as 32 base 6.
From _Omar E. Pol_, Feb 21 2010: (Start)
It appears that this can be written as a triangle :
  0,
  1,2,3,4,5,
  1,2,3,4,5,6,2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,
  1,2,3,4,5,6,2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,6,7,8,9,10,11,2...
where the rows converge to A173526.
See the conjecture in the entry A000120. (End)

Indranil Ghosh, 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. A231672, A231673, A231674, A231675, A138530.
Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, this sequence, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Cf. A173526. - Omar E. Pol, Feb 21 2010

[&+Intseq(n,6):n in [0..105]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 6], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 5}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%6,a(n-1)+1,a(n/6)))
```

a(n) = sumdigits(n, 6); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(6n+i) = a(n)+i for 0 <= i <= 5.
a(n) = n-5*(Sum_{k>0} floor(n/6^k)) = n-5*A054895(n). (End)
a(n) = A138530(n,6) for n > 5. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030567(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 6^floor(log_6(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 6*log(6)/5 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

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

A031087 Triangle T(n,k): write n in base 9, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

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

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031298 for the base-2 to base-10 analogs.

Cf. A031076, A007095.

a031087 n k = a031087_row n !! (k-1)
a031087_row n | n < 9     = [n]
              | otherwise = m : a031087_row n' where (n',m) = divMod n 9
a031087_tabf = map a031087_row [0..]
-- Reinhard Zumkeller, Jul 07 2015

A031087(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\9^k%9 \\ 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

A031045 Triangle T(n,k): write n in base 8, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Robert Israel, Table of n, a(n) for n = 0..10003 (rows 0 to 3646, flattened)

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031087, A031298 for the base-2 to base-10 analogs.

seq(op(convert(n,base,8)),n=0..100); # Robert Israel, Jul 22 2019

Flatten[Table[Reverse[IntegerDigits[n,8]],{n,80}]] (* Harvey P. Dale, Aug 08 2011 *)

A031045(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.");*/n\8^k%8 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... Note: The operation could be done using bitwise arithmetic, bitand(n>>(3*k),7), but this is not significantly faster in PARI. - M. F. Hasler, Jul 21 2013

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

A031007 Triangle T(n,k): Write n in base 7, reverse order of digits, to get row n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 6

Offset: 0

Clark Kimberling

Cf. A030308, A030341, A030386, A031235, A030567, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,7]],{n,0,50}]] (* Harvey P. Dale, Feb 25 2014 *)

A031007(n, k=-1)={k<0&&error("Flattened sequence not yet implemented.");n\7^k%7} \\ Assuming that columns start with k=0 as in A030308, A030341, ... TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

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

cp's OEIS Frontend

A030548 Write n in base 6 and juxtapose.

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A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

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

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A053827 Sum of digits of (n written in base 6).

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

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A031087 Triangle T(n,k): write n in base 9, reverse order of digits.

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A031045 Triangle T(n,k): write n in base 8, reverse order of digits.

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