cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Antti Karttunen, Nov 23 2016

Keywords

Comments

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), ...

Examples

			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

Links

Crossrefs

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).

Programs

Formula

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

Views

Author

Antti Karttunen, Nov 23 2016

Keywords

Comments

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

Links

Crossrefs

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

Programs

Formula

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

Views

Author

Antti Karttunen, Nov 23 2016

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Scheme
    ;; 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.

Formula

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

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

Views

Author

Antti Karttunen, Nov 23 2016

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Antti Karttunen, Nov 23 2016

Keywords

Links

Crossrefs

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

Programs

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

Formula

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A278539(n), and r = A278538(n).
Showing 1-5 of 5 results.

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