A264977 -id:A264977 - 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

A277701 Positions of ones in A264977; positions of twos in A277330.

Original entry on oeis.org

1, 5, 13, 29, 41, 61, 85, 125, 173, 209, 253, 281, 313, 349, 421, 509, 565, 629, 701, 845, 929, 1021, 1133, 1261, 1405, 1693, 1861, 2045, 2269, 2525, 2665, 2813, 3121, 3313, 3389, 3725, 3905, 4093, 4541, 4841, 5053, 5209, 5257, 5333, 5629, 5993, 6245, 6629, 6781, 7453, 7813, 8189, 8537, 9085, 9593, 9685, 9905, 10109, 10421, 10517, 10669, 10921

Offset: 1

Antti Karttunen, Oct 27 2016

nonn

Positions in A260443 of terms that are twice square (terms in A001105, although not all of them are present in A260443).

Antti Karttunen, Table of n, a(n) for n = 1..720

Row 1 of A277710.
Cf. A001105, A260443, A264977, A277330.
Cf. also A277712, A277713.

A277712(n) = 2*a(n) for all n >= 1.

A277713 Positions of 3's in A264977; positions of 6's in A277330.

Original entry on oeis.org

3, 39, 75, 147, 207, 291, 411, 579, 819, 927, 1155, 1635, 1851, 2307, 2487, 2583, 2919, 3267, 3699, 3903, 4611, 4971, 5163, 5835, 6531, 7395, 7803, 9219, 9939, 10323, 10839, 11667, 13059, 14787, 15603, 15999, 17895, 18435, 19875, 20295, 20643, 21675, 23331, 26115, 29571, 31203, 31995, 33327, 34383, 35787, 36867, 39747, 40587, 41283, 43347

Offset: 1

Antti Karttunen, Oct 28 2016

nonn

Positions in A260443 of terms that are six times a perfect square (terms in A033581, although not all of them are present in A260443).

All terms are multiples of three.

Antti Karttunen, Table of n, a(n) for n = 1..555

Row 3 of A277710.
Cf. A033581, A260443, A264977, A277701, A277330.
Cf. also A277701, A277712, A277714.

A277714(n) = a(n)/3.

A277712 Positions of 2's in A264977; positions of 3's in A277330.

Original entry on oeis.org

2, 10, 26, 58, 82, 122, 170, 250, 346, 418, 506, 562, 626, 698, 842, 1018, 1130, 1258, 1402, 1690, 1858, 2042, 2266, 2522, 2810, 3386, 3722, 4090, 4538, 5050, 5330, 5626, 6242, 6626, 6778, 7450, 7810, 8186, 9082, 9682, 10106, 10418, 10514, 10666, 11258, 11986, 12490, 13258, 13562, 14906, 15626, 16378, 17074, 18170, 19186, 19370, 19810

Offset: 1

Antti Karttunen, Oct 28 2016

nonn

Positions in A260443 of terms that are three times a perfect square (terms in A033428, although not all of them are present in A260443).

Row 2 of A277710.
Cf. A033428, A260443, A264977, A277701, A277330.
Cf. also A277713.

a(n) = 2*A277701(n).

A277700 a(n) = A000120(A264977(n)); number of odd terms on row n of A125184.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 4, 1, 3, 2, 3, 1, 2, 3, 3, 2, 3, 1, 4, 3, 1, 4, 5, 1, 4, 3, 3, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 3, 3, 2, 3, 3, 4, 1, 3, 4, 3, 3, 4, 1, 5, 4, 1, 5, 6, 1, 5, 4, 3, 3, 4, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 1, 2, 1, 3, 2, 1, 3, 4, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 4, 3, 3, 4, 3, 1, 2, 3, 3, 4, 3, 3, 4, 3, 3, 4, 5, 1, 4, 5, 3, 4

Offset: 0

Antti Karttunen, Oct 27 2016

nonn

Positions of even and odd terms are given by A008585 and A001651, which means that parity-wise the terms match with the Fibonacci numbers, A000045.

Antti Karttunen, Table of n, a(n) for n = 0..8193

Cf. A000120, A001221, A001222, A125184, A264977, A277330.

Cf. also A000045, A001651, A008585, A011655.

(define (A277700 n) (A000120 (A264977 n)))

a(n) = A000120(A264977(n)).
a(n) = A001221(A277330(n)) = A001222(A277330(n)).
Other identities. For all n >= 0:
a(2n) = a(n).
A000035(a(n)) = A011655(n).

A277816 a(n) = the least k > n for which A264977(k) = A264977(n), or 0 if no such k exists.

Original entry on oeis.org

0, 5, 10, 39, 20, 13, 78, 11, 40, 21, 26, 19, 156, 29, 22, 27, 80, 25, 42, 35, 52, 45, 38, 43, 312, 37, 58, 51, 44, 41, 54, 59, 160, 57, 50, 67, 84, 53, 70, 75, 104, 61, 90, 79, 76, 93, 86, 55, 624, 101, 74, 99, 116, 77, 102, 71, 88, 69, 82, 115, 108, 85, 118, 123, 320, 121, 114, 131, 100, 117, 134, 91, 168, 89, 106, 147, 140, 109, 150, 83, 208, 105

Offset: 0

Antti Karttunen, Nov 06 2016

nonn

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

Cf. A264977, A277710, A277814, A277816, A277826, A277696.
Cf. A277815 (a left inverse).
Cf. A277701, A277712, A277713, A277715 (iterates of this sequence starting from 1, 2, 3 and 9 respectively).

(define (A277816 n) (if (zero? n) n (let ((v (A264977 n))) (let loop ((k (+ 1 n))) (if (= v (A264977 k)) k (loop (+ 1 k)))))))

For all n >= 0, A277815(a(n)) = n.

A277715 Row 5 of A277710: Positions of 5's in A264977; positions of 10's in A277330.

Original entry on oeis.org

9, 21, 45, 93, 189, 381, 657, 765, 873, 1317, 1533, 1749, 2457, 2637, 3069, 3501, 4329, 4917, 5241, 5277, 5745, 6141, 6345, 7005, 8661, 9561, 9837, 10017, 10485, 10557, 11493, 12285, 12693, 14013, 15129, 17325, 17985, 19125, 19677, 20037, 20973, 21117, 21969, 22989, 24573, 25389, 26793, 28029, 30261, 31545, 34653, 35973

Offset: 1

Antti Karttunen, Oct 29 2016

nonn

Positions in A260443 of terms that are ten times a perfect square (terms in A033583, although not all of them are present in A260443).

It seems that A068156 from 9 onward is a subsequence, which (if true) would also be a sufficient condition for this sequence to be infinite.

Antti Karttunen, Table of n, a(n) for n = 1..444

Row 5 of A277710.

Cf. A033583, A068156, A260443, A264977, A277330, A277716.

A277716(n) = a(n)/3.

A277815 a(n) = the largest k < n for which A264977(k) = A264977(n), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 7, 0, 5, 0, 0, 0, 0, 0, 11, 4, 9, 14, 0, 0, 17, 10, 15, 0, 13, 0, 0, 0, 0, 0, 19, 0, 25, 22, 3, 8, 29, 18, 23, 28, 21, 0, 0, 0, 0, 34, 27, 20, 37, 30, 47, 0, 33, 26, 31, 0, 41, 0, 0, 0, 0, 0, 35, 0, 57, 38, 55, 0, 0, 50, 39, 44, 53, 6, 43, 16, 0, 58, 79, 36, 61, 46, 0, 56, 73, 42, 71, 0, 45, 0, 0, 0, 0, 0, 51

Offset: 0

Antti Karttunen, Nov 06 2016

nonn

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

Cf. A264977, A277695, A277814, A277817 (the positions of zeros).

Left inverse of A277816.

(define (A277815 n) (if (zero? n) n (let ((v (A264977 n))) (let loop ((k (- n 1))) (cond ((zero? k) 0) ((= v (A264977 k)) k) (else (loop (- k 1))))))))

For all n >= 0, a(A277816(n)) = n.

A265397 a(n) = n - A264977(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 8, 4, 0, 12, 0, 0, 0, 4, 8, 12, 16, 16, 8, 12, 0, 12, 24, 12, 0, 28, 0, 0, 0, 4, 8, 28, 16, 24, 24, 36, 32, 40, 32, 32, 16, 40, 24, 28, 0, 28, 24, 36, 48, 40, 24, 36, 0, 28, 56, 28, 0, 60, 0, 0, 0, 4, 8, 60, 16, 40, 56, 52, 32, 48, 48, 72, 48, 64, 72, 68, 64, 72, 80, 72, 64, 84, 64, 64, 32

Offset: 0

Antti Karttunen, Dec 16 2015

nonn

Note that A264977(n) gives the value of the n-th Stern polynomial (cf. A260443, A125184) evaluated at X=2 over the field GF(2), while n gives the value of the same polynomial evaluated at X=2 in the usual way.

Each term seems to be a multiple of 4.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A125184, A260443, A264977.

Cf. A023758 (positions of zeros).

Scheme
```
(define (A265397 n) (- n (A264977 n)))
```

a(n) = n - A264977(n).

A277711 a(n) = position of the first occurrence of n in A264977.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 7, 8, 81, 18, 23, 12, 17, 14, 15, 16, 153, 162, 47, 36, 49, 46, 87, 24, 73, 34, 159, 28, 33, 30, 31, 32, 177, 306, 95, 324, 97, 94, 303, 72, 137, 98, 111, 92, 297, 174, 175, 48, 145, 146, 135, 68, 1257, 318, 295, 56, 321, 66, 143, 60, 65, 62, 63, 64, 273, 354, 191, 612, 193, 190, 1119, 648, 265, 194, 1335, 188, 1233, 606

Offset: 0

Antti Karttunen, Oct 29 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A264977.
Leftmost column of A277710.
Fixed points: A023758.

(define (A277711 n) (let loop ((k 0)) (if (= (A264977 k) n) k (loop (+ 1 k))))) ;; Very crude.

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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A277701 Positions of ones in A264977; positions of twos in A277330.

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A277713 Positions of 3's in A264977; positions of 6's in A277330.

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A277712 Positions of 2's in A264977; positions of 3's in A277330.

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A277700 a(n) = A000120(A264977(n)); number of odd terms on row n of A125184.

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A277816 a(n) = the least k > n for which A264977(k) = A264977(n), or 0 if no such k exists.

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A277715 Row 5 of A277710: Positions of 5's in A264977; positions of 10's in A277330.

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A277815 a(n) = the largest k < n for which A264977(k) = A264977(n), or 0 if no such k exists.

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A265397 a(n) = n - A264977(n).

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