A140060 -id:A140060 - 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

A082447 a(n) = the number k such that s(k)=0 where s(0)=n and s(i)=s(i-1)-(s(i-1) modulo (i+1)).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 25 2003

nonn

a(n+1) = number of Mancala numbers <= n, see A007952; n occurs A028913(n-1) times consecutively. - Reinhard Zumkeller, Jun 21 2008

a(n) = number of ones <= n in A130747; see also A002491. - Reinhard Zumkeller, Jul 01 2009

			For n=4, s(0)=4, 4 ->4-4 mod 1=4 ->4-4 mod 2=4 ->4-4 mod 3=3 ->3-3 mod 4=0, hence s(4)=0 and a(4)=4.
For n=6, s(0)=6, s(1)=6-6 mod 2=6, s(2)=6-6 mod 3=6, s(3)=6-6 mod 4=6-2=4, s(4)=4-4 mod 5=0, hence a(6)=4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A073047, A140060, A140061.

Flatten@Table[First@Position[Rest@FoldList[#1-Mod[#1,#2]&,i,Range[2,i+1]],0], {i,30}] (* Birkas Gyorgy, Feb 26 2011 *)

a(n)=if(n<1, 0, s=n; c=1; while(s-s%c>0, s=s-s%c; c++); c--) \\ corrected by Dan Dima, Jan 18 2025

Conjecture: a(n) = sqrt(Pi*n) + O(1)

a(n) = A073047(n) - 1.

Name corrected by Dan Dima, Jan 18 2025

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

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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A082447 a(n) = the number k such that s(k)=0 where s(0)=n and s(i)=s(i-1)-(s(i-1) modulo (i+1)).

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A028913 First differences of A007952.

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Extensions

A140061 Triangle of quotients.

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Mathematica

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