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A173525 a(n) = 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.
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 ........
........................................................
............................................. * ........
............................................ ** ........
........................... * ............. *** ........
.......................... ** ............ **** ........
........... *............ *** ........... ***** ........
.......... ** .......... **** .......... ****** ........
......... ***.......... ***** ......... ******* ........
........ **** ........ ****** ........ ******** ........
....... ***** ....... ******* ....... ********* ........
...... ****** ...... ******** ....... ******** .........
..... ******* ...... ******* ........ ******* ..........
..... ****** ....... ****** ......... ****** ...........
..... ***** ........ ***** .......... ***** ............
..... **** ......... **** ........... **** .............
..... *** .......... *** ............ *** ..............
..... ** ........... ** ............. ** ...............
..... * ............ * .............. * ................
........................................................
..... b=7 .......... b=8 ............ b=9 ..............
... A053828 ...... A053829 ........ A053830 ............
... A173527 ...... A173528 ........ A173529 ............(End)
LINKS
FORMULA
a(n) = A053824(5^k + n - 1) where k >= ceiling(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
Sequence in context: A070671 A119281 A328943 * A070772 A094937 A215089
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 March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)