A277710 -id:A277710 - cp's OEIS frontend

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

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.

A277824 a(0) = 0; for n >= 1, a(n) = 1 + a(A277815(n)); index of the column where n is located in array A277710.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 2, 3, 2, 1, 4, 1, 1, 1, 1, 1, 4, 1, 3, 3, 2, 2, 5, 2, 2, 2, 3, 1, 1, 1, 1, 2, 3, 3, 4, 2, 2, 1, 2, 4, 2, 1, 6, 1, 1, 1, 1, 1, 5, 1, 3, 4, 3, 1, 1, 3, 3, 3, 5, 2, 3, 2, 1, 5, 4, 2, 7, 2, 1, 2, 2, 3, 4, 1, 4, 1, 1, 1, 1, 1, 4, 2, 2, 3, 5, 3, 2, 4, 5, 2, 6, 2, 1, 1, 2, 2, 3, 4, 4, 2, 2, 1

Offset: 0

Antti Karttunen, Nov 06 2016

For n >= 1, a(n) = index of the column where n is located in array A277710.

Ordinal transform of A277826 and A277884.

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

Cf. A277815, A277710.
Cf. A277817 (gives the positions of ones in this sequence).
Cf. A277826, A277884 and their scatter-plots.

(define (A277824 n) (if (zero? n) n (+ 1 (A277824 (A277815 n)))))

a(0) = 0; for n >= 1, a(n) = 1 + a(A277815(n)).

A277709 Transpose of square array A277710.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 29 2016

See A277710.

Cf. A277710.

(define (A277709 n) (A277710bi (A004736 n) (A002260 n))) ;; Code for A277710bi given in A277710.

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

Original entry on oeis.org

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)

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.

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).

A277696 Permutation of natural numbers: a(1) = 1; a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(n)).

Original entry on oeis.org

1, 2, 5, 3, 10, 7, 13, 4, 39, 15, 26, 9, 11, 18, 29, 6, 20, 92, 75, 24, 27, 49, 58, 14, 21, 16, 19, 31, 42, 62, 41, 8, 78, 33, 52, 270, 172, 196, 147, 47, 312, 56, 51, 126, 101, 143, 82, 23, 22, 34, 45, 28, 80, 32, 35, 64, 59, 96, 90, 153, 118, 95, 61, 12, 40, 224, 150, 66, 57, 129, 116, 1134, 534, 606, 316, 752, 404, 520, 207, 120, 55, 1400, 600

Offset: 1

Antti Karttunen, Nov 06 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A277817(1+n), and each right hand child as A277816(n), when the parent node contains n:
                                     1
                  ................../ \..................
                 2                                       5
       3......../ \........10                  7......../ \........13
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   4       39         15       26          9       11         18       29
  6 20   92  75     24  27   49  58      14 21   16  19     31  42   62  41
etc.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A277695.
Cf. A277710, A277816, A277817.
Cf. A277701 (the rightmost edge of the tree).

a(1) = 1; and then after, a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(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.

cp's OEIS Frontend

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

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A277824 a(0) = 0; for n >= 1, a(n) = 1 + a(A277815(n)); index of the column where n is located in array A277710.

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A277709 Transpose of square array A277710.

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

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A277696 Permutation of natural numbers: a(1) = 1; a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(n)).

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

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