A278537 -id:A278537 - cp's OEIS frontend

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

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.

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).

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).

cp's OEIS Frontend

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

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

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Programs

Scheme

Formula

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

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Programs

Scheme

Formula

A278512 Inverse permutation to A278511.

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