A099259 -id:A099259 - cp's OEIS frontend

A000960 Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate.

1, 3, 7, 13, 19, 27, 39, 49, 63, 79, 91, 109, 133, 147, 181, 207, 223, 253, 289, 307, 349, 387, 399, 459, 481, 529, 567, 613, 649, 709, 763, 807, 843, 927, 949, 1009, 1093, 1111, 1189, 1261, 1321, 1359, 1471, 1483, 1579, 1693, 1719, 1807, 1899, 1933, 2023

Offset: 1

N. J. A. Sloane

a(n) is never divisible by 2 or 5. - Thomas Anton, Nov 01 2018

			Start with
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... (A000027) and delete every second term, giving
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 ... (A005408) and delete every 3rd term, giving
1 3 7 9 13 15 19 21 25 27 ... (A056530) and delete every 4th term, giving
1 3 7 13 15 19 25 27 ... (A056531) and delete every 5th term, giving
.... Continue forever and what's left is the sequence.
(The array formed by these rows is A278492.)
For n = 5, 5^2 = 25, go down to a multiple of 4 giving 24, then to a multiple of 3 = 21, then to a multiple of 2 = 20, then to a multiple of 1 = 19, so a(5) = 19.

V. Brun, Un procédé qui ressemble au crible d'Ératosthène, Analele Stiintifice Univ. "Al. I. Cuza", Iasi, Romania, Sect. Ia Matematica, 1965, vol. 11B, pp. 47-53.
Problems 107, 116, Nord. Mat. Tidskr. 5 (1957), 114-116.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. E. Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
L. Carlitz and Chr. U. Jansen, Solution to Problem 115 and 116, Nord. Mat. Tidskr. 5 (1957), 159-161.
V. Gardiner, R.Lazarus, N. Metropolis and S. Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1955), 117-119.
H. Killingbergtro, Solution to Problem 107, Nord. Mat. Tidskr. 5 (1957), 203-205.
D. Wilson et al., Interesting sequence, SeqFan list, Nov. 2016
Index entries for sequences related to the Josephus Problem
Index entries for sequences generated by sieves

Cf. A056526, A056530, A056531, A100002, A190732, A003881.
Cf. A000012, A002491, A000959, A003309, A099259, A112557, A112558, A113742, A113743, A113744, A113745, A113746, A113747, A113748; A113749, A278484.
Cf. A119446 for triangle whose leading diagonal is A119447 and this sequence gives all possible values for A119447 (except A119447 cannot equal 1 because prime(n)/n is never 1).
Cf. A100617 (a left inverse), A100618.
Cf. A278169 (characteristic function).
Main diagonal of A278492, leftmost column of A278505, positions of zeros in A278528 & A278529.

a000960 n = a000960_list !! (n-1)
a000960_list = sieve 1 [1..] where
   sieve k (x:xs) = x : sieve (k+1) (flavius xs) where
      flavius xs = us ++ flavius vs where (u:us,vs) = splitAt (k+1) xs
-- Reinhard Zumkeller, Oct 31 2012

S[1]:={seq(i,i=1..2100)}: for n from 2 to 2100 do S[n]:=S[n-1] minus {seq(S[n-1][n*i],i=1..nops(S[n-1])/n)} od: A:=S[2100]; # Emeric Deutsch, Nov 17 2004

del[lst_, k_] := lst[[Select[Range[Length[lst]], Mod[ #, k] != 0 &]]]; For[k = 2; s = Range[2100], k <= Length[s], k++, s = del[s, k]]; s
f[n_] := Fold[ #2*Ceiling[ #1/#2 + 1] &, n, Reverse@Range[n - 1]]; Array[f, 60] (* Robert G. Wilson v, Nov 05 2005 *)

a(n)=local(A=n,D);for(i=1,n-1,D=n-i;A=D*ceil(A/D+1));return(A) \\ Paul D. Hanna, Oct 10 2005

def flavius(n):
    L = list(range(1,n+1));j=2
    while j <= len(L):
        L = [L[i] for i in range(len(L)) if (i+1)%j]
        j+=1
    return L
flavius(100)
# Robert FERREOL, Nov 08 2015

Let F(n) = number of terms <= n. Andersson, improving results of Brun, shows that F(n) = 2 sqrt(n/Pi) + O(n^(1/6)). Hence a(n) grows like Pi*n^2 / 4.
To get n-th term, start with n and successively round up to next 2 multiples of n-1, n-2, ..., 1 (compare to Mancala sequence A002491). E.g.: to get 11th term: 11->30->45->56->63->72->80->84->87->90->91; i.e., start with 11, successively round up to next 2 multiples of 10, 9, .., 1. - Paul D. Hanna, Oct 10 2005
As in Paul D. Hanna's formula, start with n^2 and successively move down to the highest multiple of n-1, n-2, etc., smaller than your current number: 121 120 117 112 105 102 100 96 93 92 91, so a(11) = 91, from moving down to multiples of 10, 9, ..., 1. - Joshua Zucker, May 20 2006
Or, similarly for n = 5, begin with 25, down to a multiple of 4 = 24, down to a multiple of 3 = 21, then to a multiple of 2 = 20 and finally to a multiple of 1 = 19, so a(5) = 19. - Joshua Zucker, May 20 2006
This formula arises in A119446; the leading term of row k of that triangle = a(prime(k)/k) from this sequence. - Joshua Zucker, May 20 2006
a(n) = 2*A073359(n-1) + 1, cf. link to posts on the SeqFan list. - M. F. Hasler, Nov 23 2016
a(n) = 1 + A278484(n-1). - Antti Karttunen, Nov 23 2016, after David W. Wilson's posting on SeqFan list Nov 22 2016

More terms and better description from Henry Bottomley, Jun 16 2000
Entry revised by N. J. A. Sloane, Nov 13 2004
More terms from Paul D. Hanna, Oct 10 2005

A100287 First occurrence of n in A100002; the least k such that A100002(k) = n.

Original entry on oeis.org

1, 2, 5, 9, 15, 25, 31, 43, 61, 67, 87, 103, 123, 139, 169, 183, 219, 241, 259, 301, 331, 361, 391, 447, 463, 511, 553, 589, 643, 687, 723, 783, 819, 867, 931, 979, 1027, 1099, 1179, 1227, 1309, 1347, 1393, 1479, 1539, 1603, 1699, 1759, 1863, 1909, 2019, 2029

Offset: 1

T. D. Noe, Nov 11 2004

nonn

Also, the first number that is crossed off at stage n in the Flavius sieve (A000960). - N. J. A. Sloane, Nov 21 2004
The sequence appears to grow roughly like 0.7825*n^2. Note that for n>2, the second occurrence of n in A100002 is at a(n)+1.
Equals main diagonal of triangle A101224, which is defined by the process starting with column 1: A101224(n,1) = n^2-n+1 for n>=1 and continuing with: A101224(n,k) = (n-k+1)*floor( (A101224(n,k-1) - 1)/(n-k+1) ) for k>1 until k=n. I.e., a(n) = A101224(n,n). - Paul D. Hanna, Dec 01 2004

Donovan Johnson, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the Josephus Problem

Cf. A000960, A099259, A100002, A101224.

Column 1 of A278507, column 2 of A278505 (without the initial 1-term).

a[n_] := Fold[#2*Ceiling[#1/#2 + 1] &, 1, Reverse@Range[n - 1]]; Array[a, 30] (* Birkas Gyorgy, Feb 16 2011 *)

{a(n)=local(A);for(k=1,n,if(k==1,A=n^2-n+1,A=(n-k+1)*floor((A-1)/(n-k+1))));A}

a(n) ~ Pi/4 * n^2 (via A000960). - Bill McEachen, Aug 08 2024

A278505 Square array constructed from Flavius sieve: Each row n (n >= 1) starts with A000960(n), followed by all numbers removed at the stage n of the sieve.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 9, 13, 8, 17, 21, 15, 19, 10, 23, 33, 37, 25, 27, 12, 29, 45, 55, 51, 31, 39, 14, 35, 57, 75, 85, 73, 43, 49, 16, 41, 69, 97, 111, 121, 99, 61, 63, 18, 47, 81, 115, 145, 159, 151, 127, 67, 79, 20, 53, 93, 135, 171, 199, 211, 193, 163, 87, 91, 22, 59, 105, 157, 205, 243, 267, 271, 247, 187, 103, 109

Offset: 1

Antti Karttunen, Nov 23 2016

The array A(row,col) is read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
   1,  2,   4,   6,   8,  10,  12,  14,  16,  18
   3,  5,  11,  17,  23,  29,  35,  41,  47,  53
   7,  9,  21,  33,  45,  57,  69,  81,  93, 105
  13, 15,  37,  55,  75,  97, 115, 135, 157, 175
  19, 25,  51,  85, 111, 145, 171, 205, 231, 265
  27, 31,  73, 121, 159, 199, 243, 283, 327, 367
  39, 43,  99, 151, 211, 267, 319, 379, 433, 487
  49, 61, 127, 193, 271, 343, 421, 483, 559, 631
  63, 67, 163, 247, 339, 427, 519, 607, 691, 793
  79, 87, 187, 303, 403, 523, 639, 739, 853, 963

Inverse: A278506.
Transpose: A278503.
Column 1: A000960, column 2: A100287 (apart from its initial 1), A099259 (differences).
Cf. A278538 (row index of n), A278539 (column index of n).
Cf. also arrays A278507 and A278511 (different variants).
Cf. also A255545 (an analogous array constructed for Lucky sieve).

(define (A278505 n) (A278505bi (A002260 n) (A004736 n)))
(define (A278505bi row col) (if (= 1 col) (A000960 row) (A278507bi row (- col 1)))) ;; Code for A278507bi given in A278507.

A(row,1) = A000960(row); for col > 1, A(row,col) = A278507(row,col-1).

For all n >= 1, A(A278538(n), A278539(n)) = n.

cp's OEIS Frontend

A000960 Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A100287 First occurrence of n in A100002; the least k such that A100002(k) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A278505 Square array constructed from Flavius sieve: Each row n (n >= 1) starts with A000960(n), followed by all numbers removed at the stage n of the sieve.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula