A278538 -id:A278538 - cp's OEIS frontend

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.

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.]

A278539 a(n) = index of the column where n is located in array A278505.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 5, 2, 6, 3, 7, 1, 8, 2, 9, 4, 10, 1, 11, 3, 12, 5, 13, 2, 14, 1, 15, 6, 16, 2, 17, 4, 18, 7, 19, 3, 20, 1, 21, 8, 22, 2, 23, 5, 24, 9, 25, 1, 26, 3, 27, 10, 28, 4, 29, 6, 30, 11, 31, 2, 32, 1, 33, 12, 34, 2, 35, 7, 36, 13, 37, 3, 38, 5, 39, 14, 40, 1, 41, 8, 42, 15, 43, 4, 44, 2, 45, 16, 46, 1, 47, 9, 48, 17, 49, 6, 50, 3, 51, 18, 52, 2, 53

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

Ordinal transform of A278538 (as the rows of A278505 are strictly growing).

Antti Karttunen, Table of n, a(n) for n = 1..10707

One more than A278529.

Cf. A278505, A278538, A278537.

Scheme
```
(define (A278539 n) (+ 1 (A278529 n)))
```

a(n) = 1 + A278529(n).

A278504 Inverse permutation to A278503.

Original entry on oeis.org

1, 3, 2, 6, 5, 10, 4, 15, 8, 21, 9, 28, 7, 36, 12, 45, 14, 55, 11, 66, 13, 78, 20, 91, 17, 105, 16, 120, 27, 136, 23, 153, 19, 171, 35, 190, 18, 210, 22, 231, 44, 253, 30, 276, 26, 300, 54, 325, 29, 351, 24, 378, 65, 406, 25, 435, 34, 465, 77, 496, 38, 528, 37, 561, 90, 595, 47, 630, 43, 666, 104, 703, 31, 741, 33, 780, 119, 820, 46

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A278503.
Cf. A278538, A278539.
Cf. also A278506, A278512.

(define (A278504 n) (let ((row (A278539 n)) (col (A278538 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A278538(n), and r = A278539(n).

A278506 Inverse permutation to A278505.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 9, 16, 8, 22, 10, 29, 14, 37, 12, 46, 15, 56, 13, 67, 17, 79, 20, 92, 21, 106, 23, 121, 27, 137, 18, 154, 30, 172, 19, 191, 28, 211, 38, 232, 35, 254, 24, 277, 47, 301, 36, 326, 26, 352, 57, 379, 25, 407, 31, 436, 68, 466, 44, 497, 45, 529, 80, 562, 54, 596, 39, 631, 93, 667, 34, 704, 32

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A278505.
Cf. A278538, A278539.
Cf. also A278504, A278512.

(define (A278506 n) (let ((row (A278538 n)) (col (A278539 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A278539(n), and r = A278538(n).

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.

A278512 Inverse permutation to A278511.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 7, 8, 10, 12, 9, 17, 11, 23, 15, 30, 13, 38, 16, 47, 14, 57, 18, 68, 21, 80, 22, 93, 24, 107, 28, 122, 19, 138, 31, 155, 20, 173, 29, 192, 39, 212, 36, 233, 25, 255, 48, 278, 37, 302, 27, 327, 58, 353, 26, 380, 32, 408, 69, 437, 45, 467, 46, 498, 81, 530, 55, 563, 40, 597, 94, 632, 35, 668, 33

Offset: 1

Antti Karttunen, Nov 23 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A278511.
Cf. A278537, A278538.
Cf. also A278504, A278506.

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A278537(n) and r = A278538(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

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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A278539 a(n) = index of the column where n is located in array A278505.

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A278504 Inverse permutation to A278503.

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A278506 Inverse permutation to A278505.

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

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A278512 Inverse permutation to A278511.

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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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