A278507 -id:A278507 - cp's OEIS frontend

A278508 Transpose of square array A278507.

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

Offset: 1

Antti Karttunen, Nov 23 2016

See A278507.

Antti Karttunen, Table of n, a(n) for n = 1..7260; the first 120 antidiagonals of the array

Transpose: A278507.

(define (A278507 n) (A278507bi (A002260 n) (A004736 n))) ;; Code for A278507bi given in A278507.

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.

A278538 a(n) = index of the row where n is located in array A278505.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

Ordinal transform of A278539 (most likely, but hinges on that also the columns of A278505 are strictly growing).

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

Cf. A000960, A100617, A278169, A278505, A278507, A278536, A278537, A278539.

(define (A278538 n) (if (not (zero? (A278169 n))) (A100617 n) (A278528 n)))

If A278169(n) = 1, a(n) = A100617(n), otherwise a(n) = A278528(n).

If n = A000960(k), a(n) = k, otherwise a(n) = number of the round in which n is removed in the Flavius sieve.

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.

A278492 Square array where row n (n >= 0) gives the numbers remaining after the n-th round of the Flavius sieve, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 3, 1, 5, 7, 7, 3, 1, 6, 9, 9, 7, 3, 1, 7, 11, 13, 13, 7, 3, 1, 8, 13, 15, 15, 13, 7, 3, 1, 9, 15, 19, 19, 19, 13, 7, 3, 1, 10, 17, 21, 25, 25, 19, 13, 7, 3, 1, 11, 19, 25, 27, 27, 27, 19, 13, 7, 3, 1, 12, 21, 27, 31, 31, 31, 27, 19, 13, 7, 3, 1, 13, 23, 31, 37, 39, 39, 39, 27, 19, 13, 7, 3, 1

Offset: 0

Antti Karttunen, Nov 23 2016, after David W. Wilson's posting on SeqFan-list Nov 22 2016

The terms of square array A(row,col) are read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left corner of the array:
1, 2, 3,  4,  5,  6,  7,  8,  9, 10 (row 0: start from A000027)
1, 3, 5,  7,  9, 11, 13, 15, 17, 19 (after the 1st round, A005408 remain)
1, 3, 7,  9, 13, 15, 19, 21, 25, 27 (after the 2nd, A047241)
1, 3, 7, 13, 15, 19, 25, 27, 31, 37
1, 3, 7, 13, 19, 25, 27, 31, 39, 43
1, 3, 7, 13, 19, 27, 31, 39, 43, 49
1, 3, 7, 13, 19, 27, 39, 43, 49, 61
1, 3, 7, 13, 19, 27, 39, 49, 61, 63
1, 3, 7, 13, 19, 27, 39, 49, 63, 67
1, 3, 7, 13, 19, 27, 39, 49, 63, 79

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of the array
D. Wilson et al., Interesting sequence, SeqFan list, Nov. 2016
Index entries for sequences generated by sieves
Index entries for sequences related to the Josephus Problem

One more than A278482.
Transpose: A278493.
Rows: A000027, A005408, A047241, A056530, A056531.
Main diagonal: A000960.
Cf. A278507 (the numbers removed at each round).
Similarly constructed arrays for other sieves: A258207, A260717.

(define (A278492 n) (A278492bi (A002262 n) (A025581 n)))
(define (A278492bi row col) (+ 1 (A278482bi row col)))

A(n,k) = 1 + A278482(n,k).

A278511 Square array constructed from Flavius sieve, shifted version, read by descending antidiagonals.

Original entry on oeis.org

2, 4, 3, 6, 5, 7, 8, 11, 9, 13, 10, 17, 21, 15, 19, 12, 23, 33, 37, 25, 27, 14, 29, 45, 55, 51, 31, 39, 16, 35, 57, 75, 85, 73, 43, 49, 18, 41, 69, 97, 111, 121, 99, 61, 63, 20, 47, 81, 115, 145, 159, 151, 127, 67, 79, 22, 53, 93, 135, 171, 199, 211, 193, 163, 87, 91, 24, 59, 105, 157, 205, 243, 267, 271, 247, 187, 103, 109

Offset: 1

Antti Karttunen, Nov 23 2016

Note how in comparison to A278505, the even numbers on the first row have been shifted one step left, "pushing" term 1 out of the array proper. This was done to obtain a better alignment with arrays like A083221 and A255127 associated with other sieves, from which one may then induce permutations by cross-referencing. (See also A255551.)

			The top left corner of the array:
   2,  4,   6,   8,  10,  12,  14,  16,  18,  20
   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

Antti Karttunen, Table of n, a(n) for n = 1..10440 [the first 144 antidiagonals of the array; offset adapted by _Georg Fischer_, Jul 15 2020]
Index entries for sequences generated by sieves
Index entries for sequences related to the Josephus Problem

Inverse: A278512.
Cf. A000960 (column 1, but with its initial 1 replaced by 2), A278505, A278507.
Cf. A278538 (row index of n), A278537 (column index of n).
Cf. A083221, A255127, A255551 (analogous arrays constructed from other sieves).

(define (A278511 n) (if (<= n 1) n (A278511bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A278511bi row col) (cond ((= 1 row) (+ col col)) ((= 1 col) (A000960 row)) (else (A278507bi row (- col 1)))))
;; Code for A278507bi given in A278507.

A(1,col) = 2*col; For row > 1, A(row,1) = A000960(row) if col = 1, otherwise, A(row,col) = A278507(row,col-1).

For all n > 1, A(A278538(n), A278537(n)) = n.

A278528 a(n) = number of the round in which n is removed in the Flavius sieve, 0 if it is never removed (when n is one of the terms of A000960).

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 3, 1, 2, 1, 0, 1, 4, 1, 2, 1, 0, 1, 3, 1, 2, 1, 5, 1, 0, 1, 2, 1, 6, 1, 3, 1, 2, 1, 4, 1, 0, 1, 2, 1, 7, 1, 3, 1, 2, 1, 0, 1, 5, 1, 2, 1, 4, 1, 3, 1, 2, 1, 8, 1, 0, 1, 2, 1, 9, 1, 3, 1, 2, 1, 6, 1, 4, 1, 2, 1, 0, 1, 3, 1, 2, 1, 5, 1, 10, 1, 2, 1, 0, 1, 3, 1, 2, 1, 4, 1, 7, 1, 2, 1, 11, 1, 3, 1, 2, 1, 0, 1, 5, 1, 2, 1, 4, 1, 3, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

a(n) = index of the row 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

Cf. A278507, A278529 (the other index), A278538.

Cf. A000960 (positions of zeros).

;; Very crude. Find it with two nested loops. (Maybe a closed form exists?)
(define (A278528 n) (cond ((not (zero? (A278169 n))) 0) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A278507bi row col) n) (if (= (A278507bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col)))))))))
;; Code for A278507bi given in A278507.

Conjecture: a(n) = [C > 0] * C where we start with A := n, B := n - 1, C, m := 0 and until A = B consecutively apply m := m + 1, C := A - B, A := abs(A - m - (A mod m)), B := abs(B - m - (B mod m)). - Mikhail Kurkov, May 19 2025

cp's OEIS Frontend

A278508 Transpose of square array A278507.

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A100287 First occurrence of n in A100002; the least k such that A100002(k) = 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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A278538 a(n) = index of the row where n is located in array A278505.

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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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A278492 Square array where row n (n >= 0) gives the numbers remaining after the n-th round of the Flavius sieve, read by descending antidiagonals.

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A278511 Square array constructed from Flavius sieve, shifted version, read by descending antidiagonals.

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A278528 a(n) = number of the round in which n is removed in the Flavius sieve, 0 if it is never removed (when n is one of the terms of A000960).

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