A277824 -id:A277824 - cp's OEIS frontend

A277710 Square array A(r,c), where each row r lists all numbers k for which A264977(k) = r, read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

1, 5, 2, 13, 10, 3, 29, 26, 39, 4, 41, 58, 75, 20, 9, 61, 82, 147, 52, 21, 6, 85, 122, 207, 116, 45, 78, 7, 125, 170, 291, 164, 93, 150, 11, 8, 173, 250, 411, 244, 189, 294, 19, 40, 81, 209, 346, 579, 340, 381, 414, 35, 104, 105, 18, 253, 418, 819, 500, 657, 582, 67, 232, 165, 42, 23, 281, 506, 927, 692, 765, 822, 131, 328, 213, 90, 43, 12

Offset: 1

Antti Karttunen, Oct 29 2016

Alternative description: Each row r lists the positions of A019565(r) in A277330.
Odd terms occur only on rows with odd index, and even terms only on rows with even index. Specifically: all terms k on row r are equal to r modulo 4, thus the first differences of each row are all multiples of 4.
All the terms on any particular row are either all multiples of two (or respectively: three, or six), or none of them are.

			The top left 12 x 12 corner of the array:
   1,   5,  13,  29,  41,   61,   85,  125,  173,  209,  253,  281
   2,  10,  26,  58,  82,  122,  170,  250,  346,  418,  506,  562
   3,  39,  75, 147, 207,  291,  411,  579,  819,  927, 1155, 1635
   4,  20,  52, 116, 164,  244,  340,  500,  692,  836, 1012, 1124
   9,  21,  45,  93, 189,  381,  657,  765,  873, 1317, 1533, 1749
   6,  78, 150, 294, 414,  582,  822, 1158, 1638, 1854, 2310, 3270
   7,  11,  19,  35,  67,  131,  259,  311,  359,  515,  619,  655
   8,  40, 104, 232, 328,  488,  680, 1000, 1384, 1672, 2024, 2248
  81, 105, 165, 213, 333,  429,  669,  861, 1341, 1725, 2685, 2721
  18,  42,  90, 186, 378,  762, 1314, 1530, 1746, 2634, 3066, 3498
  23,  43,  79,  83, 103,  155,  163,  203,  307,  323,  403,  611
  12, 156, 300, 588, 828, 1164, 1644, 2316, 3276, 3708, 4620, 6540

Transpose: A277709.
Column 1: A277711, sorted into ascending order: A277817.
Row 1: A277701, Row 2: A277712 (= 2*A277701), Row 3: A277713, Row 4: 4*A277701, Row 5: A277715, Row 6: 2*A277713. Row 8: 8*A277701, Row 10: 2*A277715.
Cf. A277824 (the index of the column where n is located in this array).
Cf. A019565, A264977, A277330, A277816 and permutation pair A277695 & A277696.

A(r,1) = A277711(r); for c > 1, A(r,c) = A277816(A(r,c-1)).
Other identities. For all r>=1, c>=1:
A(2*r,c) = 2*A(r,c).
A(r,c) modulo 4 = r modulo 4.

The dispersion-style formula added by Antti Karttunen, Nov 06 2016

A277817 Positions of zeros in A277815; sequence A277711 sorted into ascending order.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 23, 24, 28, 30, 31, 32, 33, 34, 36, 46, 47, 48, 49, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 81, 87, 92, 94, 95, 96, 97, 98, 111, 112, 120, 124, 126, 127, 128, 129, 130, 132, 135, 136, 137, 143, 144, 145, 146, 153, 159, 162, 174, 175, 177, 184, 188, 190, 191, 192, 193, 194, 196, 222

Offset: 0

Antti Karttunen, Nov 06 2016

nonn

Numbers n for which there is no such k < n that A264977(k) = A264977(n).

After zero, numbers not in the range of A277816.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A264977, A277696, A277815, A277816.
Sequence A277711 sorted into ascending order.
Positions of ones in A277824.
Cf. A277884 (a left inverse).

Other identities. For all n >= 0:

A277884(a(n)) = n.

A277826 a(n) = the least k for which A264977(k) = A264977(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 7, 12, 1, 14, 15, 16, 17, 18, 7, 4, 9, 14, 23, 24, 17, 2, 15, 28, 1, 30, 31, 32, 33, 34, 7, 36, 17, 14, 3, 8, 1, 18, 23, 28, 9, 46, 47, 48, 49, 34, 15, 4, 17, 30, 47, 56, 33, 2, 31, 60, 1, 62, 63, 64, 65, 66, 7, 68, 33, 14, 47, 72, 73, 34, 3, 28, 17, 6, 23, 16, 81, 2, 23, 36, 1, 46, 87, 56, 73, 18, 47, 92, 9

Offset: 0

Antti Karttunen, Nov 06 2016

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A264977, A277816.
Cf. A277701, A277712, A277713, A277715 (positions of 1, 2, 3 and 9 in this sequence).
Cf. A277824, A277884 and their scatter-plots.

(define (A277826 n) (let ((v (A264977 n))) (let loop ((k 0)) (if (= v (A264977 k)) k (loop (+ 1 k))))))

A277884 a(n) = A277814(A277826(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 5, 6, 7, 8, 2, 6, 9, 1, 10, 11, 12, 13, 14, 6, 4, 8, 10, 15, 16, 13, 2, 11, 17, 1, 18, 19, 20, 21, 22, 6, 23, 13, 10, 3, 7, 1, 14, 15, 17, 8, 24, 25, 26, 27, 22, 11, 4, 13, 18, 25, 28, 21, 2, 19, 29, 1, 30, 31, 32, 33, 34, 6, 35, 21, 10, 25, 36, 37, 22, 3, 17, 13, 5, 15, 12, 38, 2, 15, 23, 1, 24, 39, 28, 37, 14, 25, 40, 8, 41

Offset: 0

Antti Karttunen, Nov 06 2016

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A264977, A277814.
Left inverse of A277817.
Cf. A277701, A277712, A277713, A277715 (positions of 1, 2, 3 and 8 in this sequence).
Cf. A277824, A277826 and their scatter-plots.

(define (A277884 n) (A277814 (A277826 n)))

a(n) = A277814(A277826(n)).
Other identities. For all n >= 0:
a(A277817(n)) = n.

cp's OEIS Frontend

A277710 Square array A(r,c), where each row r lists all numbers k for which A264977(k) = r, read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A277817 Positions of zeros in A277815; sequence A277711 sorted into ascending order.

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A277826 a(n) = the least k for which A264977(k) = A264977(n).

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A277884 a(n) = A277814(A277826(n)).

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