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

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

Author

Omar E. Pol, Feb 20 2010

Keywords

• nonn
• base
• look

Comments

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)

Links

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

Crossrefs

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

Programs

Haskell

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

Maple

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) ;

Mathematica

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

PARI

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

PARI

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

Formula

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

Extensions

More terms from Vincenzo Librandi, Aug 02 2010