A082447 -id:A082447 - cp's OEIS frontend

A007952 Generated by a sieve: keep first number, drop every 2nd, keep first, drop every 3rd, keep first, drop every 4th, etc.

0, 1, 3, 5, 9, 11, 17, 21, 29, 33, 41, 47, 57, 59, 77, 81, 101, 107, 117, 131, 149, 153, 173, 191, 209, 213, 239, 257, 273, 281, 321, 329, 359, 371, 401, 417, 441, 453, 497, 509, 539, 569, 611, 621, 647, 671, 717, 731, 779, 801, 839, 869, 917, 929, 989, 1001, 1053, 1067

Offset: 0

N. J. A. Sloane, R. Muller

Also called the sieve of Tchoukaillon (or Mancala, or Kalahari).
If k+1 occurs at rank i for the first time, then i is given by the program: i = 0: for j = k to 1 step -1: i = 1 + i + int ( i / j ): next: - Claude Lenormand (claude.lenormand(AT)free.fr), Jan 15 2001
A082447(n+1) = (number of terms <= n); see A141262 for primes. - Reinhard Zumkeller, Jun 21 2008

Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11 (1957), 26-31.
M. Le, On the Smarandache n-ary Sieve, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 146-147.

L. K. Mitchell, Table of n, a(n) for n = 0..7549
D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
D. M. Broline and Daniel E. Loeb, The combinatorics of Mancala-Type games: Ayo, Tchoukaillon and 1/Pi, arXiv:math/9502225 [math.CO], 1995; J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.
P. Erdős and E. Jabotinsky, On a sequence of integers ..., Indagationes Math., 20, 115-128, 1958. part I part II
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
F. Smarandache, Only Problems, Not Solutions!
Index entries for sequences generated by sieves

Cf. A002491, A028920, A028931, A028932, A028933, A108696, A140060, A141271, A141272.

a007952 n = a007952_list !! n
a007952_list = f 1 [0..] where
   f k (x:xs) = x : f (k + 1) (g xs) where
     g ws = us ++ (g vs) where (us, _:vs) = splitAt k ws
-- Reinhard Zumkeller, Jan 19 2014

f[n_] := Fold[#2*Floor[#1/#2 + 1] &, n, Reverse@ Range[n - 1]]; Array[f, 55] (* From David Wilson *)

a(n) = my(ret=0); forstep(k=n,1,-1, ret++; ret+=(-ret)%k); ret; \\ Kevin Ryde, Sep 30 2022

Equals A002491(n) - 1. Equals A108696 - 2.

Corrected and extended by David W. Wilson

A028913 First differences of A007952.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 4, 8, 6, 10, 2, 18, 4, 20, 6, 10, 14, 18, 4, 20, 18, 18, 4, 26, 18, 16, 8, 40, 8, 30, 12, 30, 16, 24, 12, 44, 12, 30, 30, 42, 10, 26, 24, 46, 14, 48, 22, 38, 30, 48, 12, 60, 12, 52, 14, 54, 52, 26, 12, 66, 54, 60, 10, 26, 60, 60, 10, 74, 30, 52, 56, 64, 14, 34

Offset: 0

N. J. A. Sloane

nonn

R. Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A082447, A140060, A140061.

More terms from Reinhard Zumkeller, Jun 21 2008

A130747 A self-referential sequence related to Mancala solitaire (see comment).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 2, 9, 1, 10, 4, 11, 1, 12, 2, 13, 5, 14, 3, 15, 1, 16, 6, 17, 1, 18, 2, 19, 7, 20, 4, 21, 1, 22, 8, 23, 3, 24, 1, 25, 9, 26, 5, 27, 2, 28, 10, 29, 1, 30, 1, 31, 11, 32, 6, 33, 4, 34, 12, 35, 3, 36, 2, 37, 13, 38, 7, 39, 1, 40, 14, 41

Offset: 1

Benoit Cloitre, Jul 12 2007

To build the sequence, start from:
1,,2,,3,,4,,5,,6,,7,,8,,9,,10,,11,,12,,...
At the n-th step use the rule: " fill a(n)-th hole with a(n) " (holes are numbered from 1 at each step)
So step 1 is "fill first hole with 1", giving:
1,1,2,,3,,4,,5,,6,,7,,8,,9,,10,,11,,12,_,...
Since a(2)=1, step 2 is still "fill first hole with 1", giving:
1,1,2,1,3,,4,,5,,6,,7,,8,,9,,10,,11,,12,,...
Since a(3)=2, step 3 is "fill second hole with 2", giving:
1,1,2,1,3,,4,2,5,,6,,7,,8,,9,,10,,11,,12,_,...
Since a(4)=1, step 4 is "fill first hole with 1", giving:
1,1,2,1,3,1,4,2,5,,6,,7,,8,,9,,10,,11,,12,,...
Since a(5)=3, step 5 is "fill third hole with 3", giving:
1,1,2,1,3,1,4,2,5,,6,,7,3,8,,9,,10,,11,,12,_,...
Iterating the process indefinitely yields:
1,1,2,1,3,1,4,2,5,1,6,1,7,3,8,2,9,1,10,4,11,1,12,2,13,5,...
Indices where 1's occur are n=1,2,4,6,10,... which are the smallest number of stones in Mancala solitaire which make use of the n-th hole. If f(k) denotes this sequence then lim_{k->oo} k^2/f(k) = Pi.
Ordinal transform of A028920. - Benoit Cloitre, Aug 03 2007
Although A028920 and A130747 are not fractal sequences (according to Kimberling's definition) we say they are "mutual fractal sequences" since the ordinal transform of one gives the other. - Benoit Cloitre, Aug 03 2007
a(A002491(n)) = 1. - Reinhard Zumkeller, Jun 23 2009
A082447(n) = number of ones <= n. - Reinhard Zumkeller, Jul 01 2009
From Benoit Cloitre, Jul 17 2022: (Start)
Another way (less self-referent) to construct the sequence.
Step 1: Let's start from the integers separated by a hole:
1,,2,,3,,4,,5,,6,,7,,8,,9,,10,,11,,12,,...
Step 2: Put integers in the holes leaving 2 holes between each integer giving:
1,*1*,2,,3,,4,*2*,5,,6,,7,*3*,8,,9,,10,*4*,11,,12,,...
Step 3: Put integers in the holes leaving 3 holes between each integer giving:
1,1,2,*1*,3,,4,2,5,,6,,7,3,8,*2*,9,,10,4,11,,12,,...
Step 4: Put integers in the holes leaving 4 holes between each integer giving:
1,1,2,1,3,*1*,4,2,5,,6,,7,3,8,2,9,,10,4,11,,12,*2*,...
Iterating the process yields the sequence
1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 2, 9, 1, 10, 4, 11, 1, 12, 2,... (End)

Benoit Cloitre, Pi in a hole, in preparation, 2007
Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11(1957), 26-31.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.4.7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Franklin T. Adams-Watters, Doubly Fractal Sequences and ordinal transform
D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
P. Erdős and E. Jabotinsky, On sequences of integers generated by a sieving process, I, Indagationes Math., 20, 115-123, 1958.
P. Erdős and E. Jabotinsky, On sequences of integers generated by a sieving process, II, Indagationes Math., 20, 124-128, 1958.

Cf. A002491.

max = 100; A130747 = Flatten[ Transpose[ {Range[max], Table[0, {max}]}]]; Do[ hole = Last[ Position[ A130747, 0, 1, A130747[[n]] ]]; A130747[[hole]] = A130747[[n]], {n, 1, max}]; A130747 (* Jean-François Alcover, Dec 08 2011 *)

A140060 Array of quotients.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 3, 5, 4, 3, 6, 6, 6, 4, 7, 6, 6, 4, 8, 8, 6, 4, 9, 8, 6, 4, 10, 10, 9, 8, 5, 11, 10, 9, 8, 5, 12, 12, 12, 12, 10, 6, 13, 12, 12, 12, 10, 6, 14, 14, 12, 12, 10, 6, 15, 14, 12, 12, 10, 6, 16, 16, 15, 12, 10, 6, 17, 16, 15, 12, 10, 6, 18, 18, 18, 16, 15, 12, 7, 19, 18, 18

Offset: 1

Clark Kimberling, May 03 2008

1. k divides Q(n,k) for each k.

2. The numbers in row n are distinct if and only if n is a term of the Sieve of Tchoukaillon (or Mancala, or Kalahari), A007952.

			First 8 rows:
1
2 2
3 2
4 4 3
5 4 3
6 6 6 4
7 6 6 4
8 8 6 4
For row 5: Q(5,1)=5, Q(5,2)=2*[5/2]=4, Q(5,3)=3*[4/3]=3.

Cf. A140061.

For n>=1, for k=1,2,...,A082447(n), Q(n,1)=n, Q(n,k)=k*Floor(Q(n,k-1)/k).

A140061 Triangle of quotients.

Original entry on oeis.org

1, 3, 2, 5, 4, 3, 9, 8, 6, 4, 11, 10, 9, 8, 5, 17, 16, 15, 12, 10, 6, 21, 20, 18, 16, 15, 12, 7, 29, 28, 27, 24, 20, 18, 14, 8, 33, 32, 30, 28, 25, 24, 21, 16, 9, 41, 40, 39, 36, 35, 30, 28, 24, 18, 10, 47, 46, 45, 44, 40, 36, 35, 32, 27, 20, 11, 57, 56, 54, 52, 50, 48, 42, 40, 36

Offset: 1

Clark Kimberling, May 03 2008

Column 1 is essentially A007952. - Clark Kimberling, Aug 27 2008

			First 6 rows:
1
3 2
5 4 3
9 8 6 4
11 10 9 8 5
17 16 15 12 10 6

Cf. A140060.

Cf. A007952.

Flatten@Table[Reverse@FoldList[#2*Floor[#1/#2+1]&,i,Reverse@Range[i-1]],{i,10}] (* Birkas Gyorgy, Feb 26 2011 *)

The triangular subarray of A140060 consisting of rows whose terms are distinct. If n is the first term in a row (i.e., if n is a term of A082447), then Q(n,1)=n, Q(n,k)=k*Floor(Q(n,k-1)/k).

cp's OEIS Frontend

A007952 Generated by a sieve: keep first number, drop every 2nd, keep first, drop every 3rd, keep first, drop every 4th, etc.

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A028913 First differences of A007952.

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A130747 A self-referential sequence related to Mancala solitaire (see comment).

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A140060 Array of quotients.

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A140061 Triangle of quotients.

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