A277712 -id:A277712 - cp's OEIS frontend

A264977 a(0) = 0, a(1) = 1, a(2n) = 2a(n), a(2*n+1) = a(n) XOR a(n+1).

0, 1, 2, 3, 4, 1, 6, 7, 8, 5, 2, 7, 12, 1, 14, 15, 16, 13, 10, 7, 4, 5, 14, 11, 24, 13, 2, 15, 28, 1, 30, 31, 32, 29, 26, 7, 20, 13, 14, 3, 8, 1, 10, 11, 28, 5, 22, 19, 48, 21, 26, 15, 4, 13, 30, 19, 56, 29, 2, 31, 60, 1, 62, 63, 64, 61, 58, 7, 52, 29, 14, 19, 40, 25, 26, 3, 28, 13, 6, 11, 16, 9, 2, 11, 20, 1, 22

Offset: 0

Antti Karttunen, Dec 10 2015

a(n) is the n-th Stern polynomial (cf. A260443, A125184) evaluated at X = 2 over the field GF(2).

For n >= 1, a(n) gives the index of the row where n occurs in array A277710.

			In this example, binary numbers are given zero-padded to four bits.
a(2) = 2a(1) = 2 * 1 = 2.
a(3) = a(1) XOR a(2) = 1 XOR 2 = 0001 XOR 0010 = 0011 = 3.
a(4) = 2a(2) = 2 * 2 = 4.
a(5) = a(2) XOR a(3) = 2 XOR 3 = 0010 XOR 0011 = 0001 = 1.
a(6) = 2a(3) = 2 * 3 = 6.
a(7) = a(3) XOR a(4) = 3 XOR 4 = 0011 XOR 0100 = 0111 = 7.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A000225, A001511, A002487, A003987, A010060, A011655, A048675, A055396, A125184, A248663, A260443, A265397, A277330.
Cf. A023758 (the fixed points).
Cf. also A068156, A168081, A265407.
Cf. A277700 (binary weight of terms).
Cf. A277701, A277712, A277713 (positions of 1's, 2's and 3's in this sequence).
Cf. A277711 (position of the first occurrence of each n in this sequence).
Cf. also arrays A277710, A099884.

recurXOR[0] = 0; recurXOR[1] = 1; recurXOR[n_] := recurXOR[n] = If[EvenQ[n], 2recurXOR[n/2], BitXor[recurXOR[(n - 1)/2 + 1], recurXOR[(n - 1)/2]]]; Table[recurXOR[n], {n, 0, 100}] (* Jean-François Alcover, Oct 23 2016 *)

class Memoize:
    def _init_(self, func):
        self.func=func
        self.cache={}
    def _call_(self, arg):
        if arg not in self.cache:
            self.cache[arg] = self.func(arg)
        return self.cache[arg]
@Memoize
def a(n): return n if n<2 else 2*a(n//2) if n%2==0 else a((n - 1)//2)^a((n + 1)//2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 27 2017

a(0) = 0, a(1) = 1, a(2*n) = 2*a(n), a(2*n+1) = a(n) XOR a(n+1).
a(n) = A248663(A260443(n)).
a(n) = A048675(A277330(n)). - Antti Karttunen, Oct 27 2016
Other identities. For all n >= 0:
a(n) = n - A265397(n).
From Antti Karttunen, Oct 28 2016: (Start)
A000035(a(n)) = A000035(n). [Preserves the parity of n.]
A010873(a(n)) = A010873(n). [a(n) mod 4 = n mod 4.]
A001511(a(n)) = A001511(n) = A055396(A277330(n)). [In general, the 2-adic valuation of n is preserved.]
A010060(a(n)) = A011655(n).
a(n) <= n.
For n >= 2, a((2^n)+1) = (2^n) - 3.
The following two identities are so far unproved:
For n >= 2, a(3*A000225(n)) = a(A068156(n)) = 5.
For n >= 2, a(A068156(n)-2) = A062709(n) = 2^n + 3.
(End)

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.

Original entry on oeis.org

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

A277330 a(0)=1, a(1)=2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).

Original entry on oeis.org

1, 2, 3, 6, 5, 2, 15, 30, 7, 10, 3, 30, 35, 2, 105, 210, 11, 70, 21, 30, 5, 10, 105, 42, 77, 70, 3, 210, 385, 2, 1155, 2310, 13, 770, 231, 30, 55, 70, 105, 6, 7, 2, 21, 42, 385, 10, 165, 66, 143, 110, 231, 210, 5, 70, 1155, 66, 1001, 770, 3, 2310, 5005, 2, 15015, 30030, 17, 10010, 3003, 30, 715, 770, 105, 66, 91, 154, 231, 6, 385, 70, 15, 42, 11, 14, 3, 42, 55, 2

Offset: 0

Antti Karttunen, Oct 27 2016

nonn

Each term is a squarefree number, A005117.

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

Cf. A001511, A003961, A005117, A007913, A019565, A048675, A055396, A260443, A264977, A277707.
Cf. A023758 (positions where coincides with A260443).
Cf. A277701, A277712, A277713 for the positions of 2's, 3's and 6's in this sequence, which are also the first three rows of array A277710.
Cf. also A255483.

a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).
Other identities. For all n >= 0:
a(n) = A007913(A260443(n)).
a(n) = A019565(A264977(n)), A048675(a(n)) = A264977(n).
A055396(a(n)) = A277707(A260443(n)) = A001511(n).

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.

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.

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

A264977 a(0) = 0, a(1) = 1, a(2*n) = 2*a(n), a(2*n+1) = a(n) XOR a(n+1).

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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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A277330 a(0)=1, a(1)=2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).

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

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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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A264977 a(0) = 0, a(1) = 1, a(2n) = 2a(n), a(2*n+1) = a(n) XOR a(n+1).