A100287 -id:A100287 - cp's OEIS frontend

A099264 A000960(n) - A100287(n).

0, 1, 2, 4, 4, 2, 8, 6, 2, 12, 4, 6, 10, 8, 12, 24, 4, 12, 30, 6, 18, 26, 8, 12, 18, 18, 14, 24, 6, 22, 40, 24, 24, 60, 18, 30, 66, 12, 10, 34, 12, 12, 78, 4, 40, 90, 20, 48, 36, 24, 4, 132, 24, 6, 56, 40, 6, 144, 32, 20, 116, 40, 74, 40, 120, 48, 82, 66, 116, 22, 30, 96, 80, 10, 60

Offset: 1

N. J. A. Sloane, Nov 16 2004

Always positive.

Cf. A000960, A100002, A099264.

More terms from David Wasserman, Feb 27 2008

A099259 A100287(n+1) - A000960(n).

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 4, 12, 4, 8, 12, 14, 6, 22, 2, 12, 18, 6, 12, 24, 12, 4, 48, 4, 30, 24, 22, 30, 38, 14, 20, 12, 24, 4, 30, 18, 6, 68, 38, 48, 26, 34, 8, 56, 24, 6, 40, 56, 10, 86, 6, 2, 76, 42, 72, 48, 26, 4, 40, 52, 12, 54, 12, 8, 46, 26, 24, 72, 12, 42, 74, 40, 36, 98, 22, 18, 58

Offset: 1

N. J. A. Sloane, Nov 16 2004

Always positive.

Cf. A000960, A100002, A099264.

More terms from David Wasserman, Feb 27 2008

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.

A278507 Square array A(row,col) where row n lists the numbers removed in round n of Flavius sieve. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

2, 4, 5, 6, 11, 9, 8, 17, 21, 15, 10, 23, 33, 37, 25, 12, 29, 45, 55, 51, 31, 14, 35, 57, 75, 85, 73, 43, 16, 41, 69, 97, 111, 121, 99, 61, 18, 47, 81, 115, 145, 159, 151, 127, 67, 20, 53, 93, 135, 171, 199, 211, 193, 163, 87, 22, 59, 105, 157, 205, 243, 267, 271, 247, 187, 103, 24, 65, 117, 175, 231, 283, 319, 343, 339, 303, 229, 123

Offset: 1

Antti Karttunen, Nov 23 2016

			The top left corner of the array:
   2,   4,   6,   8,  10,  12,  14,  16,  18,   20
   5,  11,  17,  23,  29,  35,  41,  47,  53,   59
   9,  21,  33,  45,  57,  69,  81,  93, 105,  117
  15,  37,  55,  75,  97, 115, 135, 157, 175,  195
  25,  51,  85, 111, 145, 171, 205, 231, 265,  291
  31,  73, 121, 159, 199, 243, 283, 327, 367,  409
  43,  99, 151, 211, 267, 319, 379, 433, 487,  547
  61, 127, 193, 271, 343, 421, 483, 559, 631,  699
  67, 163, 247, 339, 427, 519, 607, 691, 793,  879
  87, 187, 303, 403, 523, 639, 739, 853, 963, 1081

Transpose: A278508.
This is array A278505 without its leftmost column, A000960.
Column 1: A100287 (apart from its initial 1).
Rows: A005843, A016969, A017629.
Cf. A278529 (column index of n), A278538 (row index of n).
Cf. A278492.
Cf. A255543 for analogous array for Lucky sieve.

(define (A278507 n) (A278507bi (A002260 n) (A004736 n)))
(define (A278507bi row col) (cond ((= 1 row) (* 2 col)) (else (A278492bi (- row 1) (+ -1 (* col (+ 1 row)))))))
;; Code for A278492bi given in A278492.

A(1,col) = 2*col; for row > 1, A(row,col) = A278492(row-1,(col*(row+1))-1). [Note that unlike this array, A278492 uses zero-based indexing for its rows and columns.]

A100002 Start with a sequence of 1's, then replace every other 1 with a 2; then replace every third of the remaining 1's with a 3 and every third of the remaining 2's with a 3; then replace every fourth remaining 1, 2 or 3 with a 4; and so on. The limiting sequence is shown here.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 1, 2, 4, 4, 3, 4, 1, 2, 5, 5, 3, 5, 1, 2, 4, 5, 3, 4, 6, 6, 1, 2, 6, 3, 7, 7, 6, 4, 7, 7, 5, 6, 1, 2, 5, 3, 8, 8, 7, 4, 8, 8, 1, 2, 6, 7, 3, 6, 5, 8, 4, 8, 5, 6, 9, 9, 1, 2, 9, 3, 10, 10, 9, 4, 10, 10, 7, 8, 9, 5, 7, 10, 1, 2, 9, 7, 3, 4, 9, 6, 11, 11, 10, 11

Offset: 1

David A. Madore, Oct 25 2004

The position of the 1's is given by A000960. - T. D. Noe, Oct 26 2004

			Here are the first 6 stages in the construction:
  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1...
  1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2...
  1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3...
  1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3...
  1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 1 2 1 2 3 3...
  1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 6 6 1 2 6 3...
  ...

T. D. Noe, Table of n, a(n) for n = 1..10000
T. D. Noe, Plot of first 5000 terms
A post on sci.math.research newsgroup.

Cf. A100287 (first occurrence of n).

#define MAXVAL 2048 /* Large enough... */
unsigned int counts[MAXVAL][MAXVAL]; /* Initialized at all 0 */ unsigned int seq_value (void) /* Successive calls return values in the sequence, in order. */ { unsigned int value; unsigned int i; value = 1; for ( i=2; i= i ) { counts[i][value] = 0; value = i; } return value; }

nn=100; t=Table[1, {nn}]; done=False; k=1; While[ !done, k++; cnt=Table[0, {k-1}]; Do[If[t[[i]]T. D. Noe *)
a[n_] := Fold[Function[{b1, b2},Fold[Function[{a1, a2},ReplacePart[a1, Pick[Position[a1, a2], Take[Flatten[Array[{Array[0 &, b2 - 1], 1} &, Length[a1]]], Length[Position[a1, a2]]], 1] -> b2]], b1, Range[b2 - 1]]], Array[1 &, n], Range[2, 2 Sqrt[n/Pi] + 1]]; a[100] (* Birkas Gyorgy, Feb 06 2011 *)

a(1, j)=1 for all j>=1; a(n, j)=a(n-1, j) except when #{i<=j s.t. a(n-1, i)=a(n-1, j)} is multiple of n, in which case a(n, j)=n; a(j) is the limit of the (stationary) a(n, j) when n tends to infinity.
It appears that the maximal value among the first n terms grows like sqrt(4n/3).
Note that the first occurrence of n is bounded by A000960; that is, A100287(n) <= A000960(n), with equality only for n=1. - T. D. Noe, Nov 12 2004

A278529 a(n) = one-based position in the round in which n is removed in the Flavius sieve, 0 if n is never removed (when n is one of the terms of A000960).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 4, 1, 5, 2, 6, 0, 7, 1, 8, 3, 9, 0, 10, 2, 11, 4, 12, 1, 13, 0, 14, 5, 15, 1, 16, 3, 17, 6, 18, 2, 19, 0, 20, 7, 21, 1, 22, 4, 23, 8, 24, 0, 25, 2, 26, 9, 27, 3, 28, 5, 29, 10, 30, 1, 31, 0, 32, 11, 33, 1, 34, 6, 35, 12, 36, 2, 37, 4, 38, 13, 39, 0, 40, 7, 41, 14, 42, 3, 43, 1, 44, 15, 45, 0, 46, 8, 47, 16, 48, 5, 49, 2, 50, 17, 51, 1, 52

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

a(n) = index of the column where n is located in array A278507, 0 if n does not occur there (when n is one of the terms of A000960).

Antti Karttunen, Table of n, a(n) for n = 1..10707
Index entries for sequences related to the Josephus Problem

One less than A278539.
Cf. A278507, A278528 (the other index).
Cf. A000960 (positions of zeros), A100287 (positions of 1's, after the initial 1).

;; Very crude. Find it with two nested loops. (Maybe a closed form exists?)
(define (A278529 n) (cond ((not (zero? (A278169 n))) 0) ((even? n) (/ n 2)) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A278507bi row col) n) (if (= (A278507bi row col) n) col (searchrow (+ 1 row)))) (else (searchcol (+ 1 col)))))))))
;; Code for A278507bi given in A278507.

For n > 1, a(A100287(n)) = 1.

A344009 The Tchoukaillon array of order infinity read by downward antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 9, 13, 10, 11, 14, 15, 19, 12, 16, 17, 21, 25, 27, 18, 20, 23, 26, 29, 31, 39, 22, 24, 28, 33, 37, 41, 43, 49, 30, 32, 35, 38, 44, 45, 51, 61, 63, 34, 36, 40, 47, 50, 55, 62, 65, 67, 79, 42, 46, 52, 53, 57, 64, 69, 73, 85, 87, 91, 48, 54, 56, 66, 68, 74, 75, 86, 89, 99, 103, 109

Offset: 1

N. J. A. Sloane, Jun 09 2021, based on an email from Don Knuth, Jun 08 2021

			The array begins:
1, 2, 4, 6, 10, 12, 18, 22, 30, 34, 42, 48, 58, 60, 78, 82, 102, 108, 118, 132, 150, 154, 174, 192, 210, 214, 240, 258, 274, 282, 322, 330, ...
3, 5, 8, 11, 16, 20, 24, 32, 36, 46, 54, 59, 72, 80, 90, 106, 114, 120, 142, 152, 168, 180, 198, 212, 228, 252, 270, 276, 318, 324, 334, ...
7, 9, 14, 17, 23, 28, 35, 40, 52, 56, 70, 76, 84, 94, 112, 116, 138, 144, 162, 172, 196, 202, 222, 234, 262, 272, 298, 320, 332, 342, ...
13, 15, 21, 26, 33, 38, 47, 53, 66, 71, 83, 92, 107, 113, 130, 140, 156, 166, 190, 200, 216, 232, 256, 264, 288, 312, 328, 336, 378, ...
19, 25, 29, 37, 44, 50, 57, 68, 77, 88, 96, 110, 119, 136, 148, 160, 178, 197, 204, 226, 250, 260, 275, 294, 323, 335, 358, 380, ...
27, 31, 41, 45, 55, 64, 74, 81, 95, 100, 117, 126, 143, 155, 167, 179, 203, 208, 238, 251, 268, 286, 316, 326, 354, 359, 406, ...
39, 43, 51, 62, 69, 75, 93, 98, 115, 124, 137, 153, 164, 176, 201, 206, 236, 239, 263, 280, 300, 317, 348, 356, 392, 407,
...
49, 61, 65, 73, 86, 97, 104, 122, 131, 146, 158, 173, 191, 205, 215, 237, 257, 278, 292, 310, 333, 352, 366, 400, 417, ...
...
The initial antidiagonals are:
[1],
[2, 3],
[4, 5, 7],
[6, 8, 9, 13],
[10, 11, 14, 15, 19],
[12, 16, 17, 21, 25, 27],
[18, 20, 23, 26, 29, 31, 39],
[22, 24, 28, 33, 37, 41, 43, 49],
[30, 32, 35, 38, 44, 45, 51, 61, 63],
[34, 36, 40, 47, 50, 55, 62, 65, 67, 79],
[42, 46, 52, 53, 57, 64, 69, 73, 85, 87, 91],
[48, 54, 56, 66, 68, 74, 75, 86, 89, 99, 103, 109],
[58, 59, 70, 71, 77, 81, 93, 97, 101, 105, 121, 123, 133],
[60, 72, 76, 83, 88, 95, 98, 104, 111, 125, 127, 135, 139, 147],
...

D. E. Knuth, Bipartite Matching, The Art of Computer Programming, Vol. 4, Pre-fascicle 14A, June 8, 2021, http://cs.stanford.edu/~knuth/fasc14a.ps.gz. See Sect. 7.5.1, Exercise 13.

D. E. Knuth, The first 32 rows

Rows: A002491, A344010, A344011, ...

Columns: A000960, A100287, A344012, ...

A101224 Triangle, read by rows, where T(n,1) = n^2-n+1 for n>=1 and T(n,k) = (n-k+1)*floor( (T(n,k-1)-1)/(n-k+1) ) for 1A000960).

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 13, 12, 10, 9, 21, 20, 18, 16, 15, 31, 30, 28, 27, 26, 25, 43, 42, 40, 36, 33, 32, 31, 57, 56, 54, 50, 48, 45, 44, 43, 73, 72, 70, 66, 65, 64, 63, 62, 61, 91, 90, 88, 84, 78, 75, 72, 69, 68, 67, 111, 110, 108, 104, 98, 96, 95, 92, 90, 88, 87, 133, 132, 130, 126

Offset: 1

Paul D. Hanna, Dec 01 2004

A variant of triangle A100452. The main diagonal equals A100287, the first number that is crossed off at stage n in the Flavius sieve (A000960). Row sums are A101105.

			T(4,4) = 9 since we start with T(4,1)=4^2-4+1=13 and then
T(4,2)=(4-2+1)*floor((T(4,1)-1)/(4-2+1))=3*floor((13-1)/3)=12,
T(4,3)=(4-3+1)*floor((T(4,2)-1)/(4-3+1))=2*floor((12-1)/2)=10,
T(4,4)=(4-4+1)*floor((T(4,3)-1)/(4-4+1))=1*floor((10-1)/1)=9.
Rows begin:
[1],
[3,2],
[7,6,5],
[13,12,10,9],
[21,20,18,16,15],
[31,30,28,27,26,25],
[43,42,40,36,33,32,31],
[57,56,54,50,48,45,44,43],
[73,72,70,66,65,64,63,62,61],...

Index entries for sequences related to the Josephus Problem

Cf. A100452, A100287, A000960, A101105.

T(n,k)=if(k==1,n^2-n+1,(n-k+1)*floor((T(n,k-1)-1)/(n-k+1)))

T(n, n) = A100287(n).

A272800 Flavius Josephus factor of n.

Original entry on oeis.org

0, 2, 0, 2, 3, 2, 0, 2, 4, 2, 3, 2, 0, 2, 5, 2, 3, 2, 0, 2, 4, 2, 3, 2, 6, 2, 0, 2, 3, 2, 7, 2, 4, 2, 3, 2, 5, 2, 0, 2, 3, 2, 8, 2, 4, 2, 3, 2, 0, 2, 6, 2, 3, 2, 5, 2, 4, 2, 3, 2, 9, 2, 0, 2, 3, 2, 10, 2, 4, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0, 2, 4, 2, 3, 2, 6, 2

Offset: 1

Max Barrentine, May 06 2016

nonn

This sequence is analogous to the smallest prime factor of n. If n is a member of A000960, a(n) = 0, otherwise a(n) = the (k+1)-st step that rejects n from Flavius Josephus' sieve.

The n-values of records of this sequence are given by A100287 (see 2004 comment by Sloane).

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

Cf. A000960, A100002, A100287.

# Function that returns an array of the first n terms.
def A272800(n):
    A, B, k = [0]*n, range(n), 1
    while kDanny Rorabaugh, May 13 2016

A344012 Column 3 of array in A344009.

Original entry on oeis.org

4, 8, 14, 21, 29, 41, 51, 65, 85, 99, 121, 135, 163, 182, 211, 229, 255, 291, 313, 351, 389, 423, 461, 507, 543, 571, 627, 673, 721, 781

Offset: 1

N. J. A. Sloane, Jun 09 2021

Cf. A000960, A100287, A344009.

cp's OEIS Frontend

A099264 A000960(n) - A100287(n).

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A099259 A100287(n+1) - A000960(n).

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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.

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A278507 Square array A(row,col) where row n lists the numbers removed in round n of Flavius sieve. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A100002 Start with a sequence of 1's, then replace every other 1 with a 2; then replace every third of the remaining 1's with a 3 and every third of the remaining 2's with a 3; then replace every fourth remaining 1, 2 or 3 with a 4; and so on. The limiting sequence is shown here.

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C

Mathematica

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A278529 a(n) = one-based position in the round in which n is removed in the Flavius sieve, 0 if n is never removed (when n is one of the terms of A000960).

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A344009 The Tchoukaillon array of order infinity read by downward antidiagonals.

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A101224 Triangle, read by rows, where T(n,1) = n^2-n+1 for n>=1 and T(n,k) = (n-k+1)*floor( (T(n,k-1)-1)/(n-k+1) ) for 1A000960).

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PARI

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A272800 Flavius Josephus factor of n.

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