A173525 -id:A173525 - cp's OEIS frontend

A053824 Sum of digits of (n written in base 5).

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

Offset: 0

Henry Bottomley, Mar 28 2000

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

			a(20) = 4 + 0 = 4 because 20 is written as 40 in base 5.
From _Omar E. Pol_, Feb 21 2010: (Start)
It appears that this can be written as a triangle:
  0,
  1,2,3,4,
  1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,
  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,...
See the conjecture in the entry A000120. (End)

Tanar Ulric, Table of n, a(n) for n = 0..10000 (terms 0..3125=5^5 from Reinhard Zumkeller).
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. A231668, A231669, A231670, A231671, A138530.
Sum of digits of n written in bases 2-16: A000120, A053735, A053737, this sequence, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Cf. A173525. - Omar E. Pol, Feb 21 2010
Cf. A173670 (last nonzero decimal digit of (10^n)!). - Washington Bomfim, Jan 01 2011

a053824 0 = 0
a053824 x = a053824 x' + d  where (x', d) = divMod x 5
-- Reinhard Zumkeller, Jan 31 2014

[&+Intseq(n, 5):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 5], {n, 0, 100}] (* or *)
Nest[Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 4}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
f[n_] := n - 4 Sum[Floor[n/5^k], {k, n}]; Array[f, 103, 0]

PARI
```
a(n)=if(n<1,0,if(n%5,a(n-1)+1,a(n/5)))
```

a(n) = sumdigits(n, 5); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(5n+i) = a(n) + i for 0 <= i <= 4;
a(n) = n - 4*Sum_{k>=1} floor(n/5^k) = n - 4*A027868(n). (End)
a(n) = A138530(n,5) for n > 4. - Reinhard Zumkeller, Mar 26 2008
If i >= 2, a(2^i) mod 4 = 0. - Washington Bomfim, Jan 01 2011
a(n) = Sum_{k>=0} A031235(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 5^floor(log_5(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 5*log(5)/4 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A173523 1+A053735(n-1), where A053735 is the sum-of-digits function in base 3.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2010

A053735 can be obtained as 0 followed by the first 2 terms of this sequence, followed by the first 6 terms, followed by the first 18 terms, ..., followed by the first 2*3^n terms, etc.
Similar observations are possible for: A063787 (base-2 case), and generic comments have been gathered in A173525 (base-5 case).
Fixed point of morphism 1->123, 2->234, 3->345 etc. (start with 1).

			If written as a triangle, begins:
1,
2,3,
2,3,4,3,4,5,
2,3,4,3,4,5,4,5,6,3,4,5,4,5,6,5,6,7,
2,3,4,3,4,5,4,5,6,3,4,5,4,5,6,5,6,7,4,5,6,5,6,7,6,7,8,...

Cf. A000120, A053735, A063787, A173524 - A173529.

A173524 a(n) = A053737(4^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053737.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2010

nonn

It appears that if A053737 is written as a triangle then the rows are initial segments of the present sequence; see the conjecture in A000120.

The comments in A173525 (base b=5 there) apply here with base b=4. The base b=3 is considered in A173523.

Cf. A000120, A053737, A063787.

Cf. A173523, A173525, A173526, A173527, A173528, A173529.

A053737 := proc(n) add(d, d=convert(n,base,4)) ; end proc:
A173524 := proc(n) local b; b := 4 ; if n < b then n; else k := n/(b-1); k := ceil(log(k)/log(b)) ; A053737(b^k+n-1) ; end if; end proc:
seq(A173524(n),n=1..100) ; # R. J. Mathar, Dec 09 2010

a(n) = A053737(4^k+n-1) where k >= ceiling(log_4(n/3)). [R. J. Mathar, Dec 09 2010]

Conjecture: Fixed point of the morphism 1->{1,2,3,...,b}, 2->{2,3,4,...,b+1}, j->{j,j+1,...,j+b-1} for b=4. [Joerg Arndt, Dec 08 2010]

A173528 a(n) = 1 + sum of digits of n-1 written in base 8.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2010

If A053829 is written as a triangle then the rows converge to this sequence; see the conjecture in A000120.

The sequence is the base b=8 case in a family of 8 sequences for base b=2 (A063787) and bases 3 to 9 (A173523 to A173529). Common aspects (recurrences etc.) of these are documented in A173525.

Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A000120, A053829, A063787, A173523 - A173529.

a053829:=func< n | &+Intseq(n, 8) >; a173528:=func< n | a053829(n-1)+1 >; [ a173528(n): n in [1..90] ]; // Klaus Brockhaus, Dec 07 2010

A173528 = lambda n: 1+sum((n-1).digits(base=8)) # D. S. McNeil, Dec 07 2010

a(n) = A053829(n-1)+1.

More terms from Vincenzo Librandi, Feb 21 2010

Definition and formula added by M. F. Hasler, Dec 06 2010

A173529 a(n) = 1 + A053830(n-1), where A053830 is the sum of the digits of its argument in base 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4

Offset: 1

Omar E. Pol, Feb 23 2010

If A053830 is regarded as a triangle then the rows converge to this sequence.

This is the case b=9 of a non-periodic sequence, with key formulas and definitions provided with b=5 in A173525. Case b=2 is in A063787, and cases b=3 to 8 are in A173523 to A173528.

Cf. A000120, A053830, A063787, A173523, A173524, A173525, A173526, A173527, A173528.

Table[1 + Plus@@IntegerDigits[n - 1, 9], {n, 90}] (* Vincenzo Librandi, Jul 01 2019 *)

a(n) = A053830(9^k + n - 1) where k >= ceiling(log_9(n/8)). - R. J. Mathar, Dec 09 2010

A173526 a(n) = 1 + A053827(n-1), where A053827 is the sum-of-digits function in base 6.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2010

If A053827 is regarded as a triangle then the rows converge to this sequence, i.e., a(n) = A053827(6^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053827.
See conjecture in the entry A000120.
This sequence is the base b=6 case equivalent to A063787 (b=2), A173523 (b=3), A173524 (b=4), A173525 (b=5). Generic comments concerning the various bases are in A173525.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000120, A053827, A063787.

Cf. A173523, A173524, A173525, A173527, A173528, A173529.

Table[1 + Total[IntegerDigits[n-1, 6]], {n, 1, 110}] (* G. C. Greubel, Jul 02 2019 *)

A053827(n)= if(n<1, 0, if(n%6, a(n-1)+1, a(n/6)));
vector(110, n, 1+A053827(n-1)) \\ G. C. Greubel, Jul 02 2019

a(n) = A053827(6^k+n-1) where k >= ceiling(log_6(n/5)). - R. J. Mathar, Dec 09 2010
Conjecture: Fixed point of the morphism 1->{1,2,3,...,b}, 2->{2,3,4,...,b+1},
j->{j,j+1,...,j+b-1} for b=6. - Joerg Arndt, Dec 08 2010

More terms from Vincenzo Librandi, Aug 02 2010

A173527 a(n) = 1 + A053828(n-1), where A053828 is the sum of digits in base 7.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2010

If A053828 is regarded as a triangle then the rows converge to this sequence, i.e., a(n) = A053828(7^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053828.
See conjecture in the entry A000120.
This is the case for base b=7 for the sum of digits. A063787 and A173523 to A173526 deal with the bases 2 to 6. A173525 contains generic remarks concerning these 8 sequences which look in equivalent ways at their sum of digits as a sequence with triangular structure.

Cf. A000120, A053828, A063787.

Cf. A173523, A173524, A173525, A173526, A173528, A173529.

A053828 := proc(n) add(d, d=convert(n,base,7)) ; end proc:
A173527 := proc(n) local b; b := 7 ; if n < b then n; else k := n/(b-1); k := ceil(log(k)/log(b)) ; A053828(b^k+n-1) ; end if; end proc:
seq(A173527(n),n=1..100) ; # R. J. Mathar, Dec 09 2010

Table[Total[IntegerDigits[n-1,7]]+1,{n,110}] (* Harvey P. Dale, Apr 01 2018 *)

a(n) = A053828(7^k+n-1) where k >= ceiling(log_7(n/6)). [R. J. Mathar, Dec 09 2010]

Conjecture: Fixed point of the morphism 1->{1,2,3,...b}, 2->{2,3,4...,b+1}, j->{j,j+1,...,j+b-1} for b=7. [Joerg Arndt, Dec 08 2010]

More terms from Vincenzo Librandi, Feb 21 2010

cp's OEIS Frontend

A053824 Sum of digits of (n written in base 5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Formula

A173523 1+A053735(n-1), where A053735 is the sum-of-digits function in base 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

A173524 a(n) = A053737(4^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053737.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

A173528 a(n) = 1 + sum of digits of n-1 written in base 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Sage

Formula

Extensions

A173529 a(n) = 1 + A053830(n-1), where A053830 is the sum of the digits of its argument in base 9.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A173526 a(n) = 1 + A053827(n-1), where A053827 is the sum-of-digits function in base 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A173527 a(n) = 1 + A053828(n-1), where A053828 is the sum of digits in base 7.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions