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A173525 1+A053824(n-1), where A053824 = sum of digits in base 5. 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, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

Comments 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 saw-tooth structure of periodicity b is added algebraically on top of a saw-tooth 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

Comment  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

FORMULA

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

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 *)

PROG

(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

(Haskell)

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

CROSSREFS

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

Sequence in context: A212176 A070671 A119281 * A070772 A094937 A215089

Adjacent sequences:  A173522 A173523 A173524 * A173526 A173527 A173528

KEYWORD

nonn,base,look

AUTHOR

Omar E. Pol, Feb 20 2010

EXTENSIONS

More terms from Vincenzo Librandi, Aug 02 2010

STATUS

approved

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Last modified February 20 06:45 EST 2018. Contains 299358 sequences. (Running on oeis4.)