A065621 -id:A065621 - cp's OEIS frontend

A246164 Permutation of natural numbers: a(1) = 1, a(A065621(n)) = A014580(a(n-1)), a(A048724(n)) = A091242(a(n)), where A065621(n) and A048724(n) are the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

1, 2, 4, 11, 8, 5, 3, 7, 6, 9, 13, 17, 47, 31, 14, 61, 21, 42, 185, 24, 87, 319, 62, 12, 25, 19, 10, 59, 20, 15, 37, 229, 49, 22, 67, 76, 415, 103, 28, 18, 55, 137, 34, 41, 16, 27, 97, 78, 425, 109, 29, 1627, 222, 54, 283, 433, 79, 373, 3053, 33, 131, 647, 108, 847, 133, 745, 6943, 44, 193, 1053, 160, 504, 4333, 587, 99

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A065621/A048724 and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).
The former are themselves permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246162.
For the comments about the cycle structure, please see A246163.

Inverse: A246163.
Similar or related permutations: A246206, A246202, A193231, A245702, A234025, A246162, A234612, A246204.
Cf. A010060, A014580, A048724, A065620, A065621, A091242, A246159, A246160.

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. when n is an odious number], a(n) = A014580(a(A065620(n)-1)), otherwise a(n) = A091242(a(- (A065620(n)))). [A065620 Converts sum of powers of 2 in binary representation of n to an alternating sum].
As a composition of related permutations:
a(n) = A246202(A193231(n)).
a(n) = A245702(A234025(n)).
a(n) = A246162(A234612(n)).
a(n) = A193231(A246204(A193231(n))).
For all n > 1, A091225(a(n)) = A010060(n). [Maps odious numbers to binary representations of irreducible GF(2) polynomials (A014580) and evil numbers to the corresponding representations of reducible polynomials (A091242), in some order. A246162 has the same property].

A379121 Odd squares k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

Original entry on oeis.org

225, 3025, 3249, 12321, 29241, 38025, 91809, 216225, 247009, 354025, 408321, 751689, 772641, 855625, 919681, 1366561, 1595169, 3814209, 9828225, 11189025, 12173121, 12709225, 29430625, 47927929, 52403121, 66471409, 67486225, 77457601, 80263681, 94148209, 100661089, 110397049, 126540001, 204232681, 264875625, 328878225

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

Of the first 2025 terms, only two, a(520) and a(1087) have multiple solutions. See the examples.

A114390 a(n) = A065621(n^2).

Original entry on oeis.org

1, 4, 25, 16, 41, 100, 81, 64, 241, 164, 137, 400, 505, 324, 289, 256, 865, 964, 953, 656, 713, 548, 1585, 1600, 1681, 2020, 1897, 1296, 1497, 1156, 1089, 1024, 3265, 3460, 3417, 3856, 4073, 3812, 3601, 2624, 2993, 2852, 2377, 2192, 2105, 6340

Offset: 1

Antti Karttunen, Feb 07 2006

nonn

Cf. A000290, A065621.

A114391 gives the positions where a(n) is square, A114392 gives the corresponding values (squares) and A114393 gives their square roots.

def A114390(n): return (m:=n**2)^ (m&~-m)<<1 # Chai Wah Wu, Jun 29 2022

a(n) = A065621(A000290(n)).

A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 128, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 384, 25, 26, 2187, 28, 29, 30, 31, 8192, 33, 34, 35, 36, 37, 38, 39, 640, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 4374, 55, 896, 57, 58, 59, 60, 61, 62, 63, 16384, 65, 66, 67, 68, 69, 70, 71, 1152, 73, 74, 75

Offset: 1

Antti Karttunen, May 27 2016

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A020639, A028234, A065621, A067029.
Cf. A270428 (same sequence sorted into ascending order).
Cf. also A270418, A270419, A270437 and permutation A273671.

f[p_, e_] := p^BitXor[e - 1, 2*e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

Multiplicative with a(p^e) = p^A065621(e).
a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
A270418(a(n)) = n, A270419(a(n)) = 1.

A284270 Square array A(r,c) = A048720(A065621(r), c) mod r, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 3, 4, 4, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 2, 1, 0, 7, 0, 0, 2, 0, 1, 0, 2, 0, 5, 6, 0, 0, 0, 0, 0, 0, 4, 0, 7, 2, 9, 0, 0, 0, 0, 4, 0, 2, 0, 1, 6, 7, 4, 0, 0, 1, 0, 1, 4, 0, 0, 8, 4, 0, 8, 8, 0, 0, 0, 0, 4, 0, 4, 0, 5, 4, 3, 0, 3, 8, 0, 0, 2, 0, 2, 0, 6, 0, 7, 2, 0, 4, 11, 2, 4

Offset: 1

Antti Karttunen, Apr 10 2017

			The top left 17 x 19 corner of the array:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   1,  2,  0,  1,  0,  0,  0,  2,  0,  0,  1,  0,  2,  0,  0,  1,  2
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   3,  1,  3,  2,  2,  1,  0,  4,  1,  4,  2,  2,  1,  0,  0,  3,  1
   2,  4,  0,  2,  0,  0,  0,  4,  0,  0,  2,  0,  4,  0,  0,  2,  4
   4,  1,  1,  2,  4,  2,  0,  4,  6,  1,  6,  4,  1,  0,  0,  1,  5
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   7,  5,  7,  1,  8,  5,  7,  2,  2,  7,  2,  1,  1,  5,  0,  4,  6
   6,  2,  6,  4,  4,  2,  0,  8,  2,  8,  4,  4,  2,  0,  0,  6,  2
   9,  7,  0,  3,  0,  0,  5,  6,  0,  0,  8,  0,  1, 10,  0,  1,  0
   4,  8,  0,  4,  0,  0,  0,  8,  0,  0,  4,  0,  8,  0,  0,  4,  8
   8,  3, 11,  6,  0,  9,  3, 12,  7,  0,  8,  5, 12,  6,  0, 11,  0
   8,  2,  2,  4,  8,  4,  0,  8, 12,  2, 12,  8,  2,  0,  0,  2, 10
   4,  8,  8,  1,  5,  1,  1,  2,  4, 10,  8,  2,  4,  2,  0,  4,  6
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  15, 13, 15,  9,  7, 13, 15,  1, 16, 14, 16,  9,  7, 13, 15,  2,  2
  14, 10, 14,  2, 16, 10, 14,  4,  4, 14,  4,  2,  2, 10,  0,  8, 12
  17, 15, 13, 11,  7,  7,  0,  3,  0, 14,  6, 14, 16,  0, 13,  6,  3

Cf. A048720, A065621, A115872, A277320, A284269 (transpose), A284273 (main diagonal), A284552 (column 1).

Row 3: A284557.

(define (A284270 n) (A284270bi (A002260 n) (A004736 n)))
(define (A284270bi row col) (modulo (A277320bi row col) row)) ;; For A277320bi, see in A277320.

A(r,c) = A277320(r,c) mod r = A048720(A065621(r), c) mod r.

A325570 Numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 131, 137, 139, 141, 143, 145, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 175, 177, 179, 181

Offset: 1

Antti Karttunen, May 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048720, A065621, A115872, A115873.
Positions of ones in A325565 and A325566.
Cf. A065091 (a subsequence), A325571 (the composite terms), A325572 (complement).
Subsequence of A005408 (odd numbers).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325570(n) = fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d==n)));

A379114 Numbers k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 35, 39, 40, 42, 44, 46, 48, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 75, 76, 77, 78, 80, 84, 86, 87, 88, 91, 92, 93, 94, 96, 100, 102, 104, 105, 108, 111, 112, 114, 115, 116, 118, 119, 120, 122, 123, 124, 126, 129, 132, 133, 136, 138, 140, 141, 143, 144

Offset: 1

Antti Karttunen, Dec 17 2024

nonn

Positions of terms > 1 in A379113.
Cf. A000396 (subsequence, at least the even terms are), A379118 (characteristic function).
Cf. also A325638, A325639 (not subsequences).

PARI
```
is_A379114(n) = (A379113(n)>1);
```

A245471 If n is odd, then a(n) = A065621(n+1). If n is even, then a(n) = n/2.

Original entry on oeis.org

2, 1, 4, 2, 14, 3, 8, 4, 26, 5, 28, 6, 22, 7, 16, 8, 50, 9, 52, 10, 62, 11, 56, 12, 42, 13, 44, 14, 38, 15, 32, 16, 98, 17, 100, 18, 110, 19, 104, 20, 122, 21, 124, 22, 118, 23, 112, 24, 82, 25, 84, 26, 94, 27, 88, 28, 74, 29, 76, 30, 70, 31, 64, 32, 194, 33, 196, 34, 206, 35, 200, 36, 218, 37, 220, 38, 214, 39, 208, 40, 242, 41, 244, 42, 254, 43, 248, 44, 234, 45, 236, 46, 230, 47, 224, 48, 162, 49, 164, 50, 174, 51, 168, 52, 186, 53, 188, 54, 182, 55, 176, 56, 146, 57, 148, 58, 158, 59, 152, 60

Offset: 1

Reinhard Muehlfeld, Jul 23 2014

A Collatz-like function: the difference is that for odd n the term 3n+1 is calculated without overflow, only using xor operations (n xor(2n+1)). It is known that for each argument the iterated function always ends up in a cycle which contains 1 (namely 1-2-1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..8192

Cf. A006370, A065621.

import Data.List (transpose)
a245471 n = a245471_list !! (n-1)
a245471_list = concat $ transpose [odds a065621_list, [1..]]
   where odds [] = []; odds [x] = []; odds (_:x:xs) = x : odds xs
-- Reinhard Zumkeller, Jul 27 2014

def A245471(n): return (m:=n+1)^ (m&~-m)<<1 if n&1 else n>>1 # Chai Wah Wu, Jun 29 2022

Definition corrected by Chai Wah Wu, Jun 29 2022

A246160 Inverse function to the injection A065621.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 0, 3, 8, 0, 0, 7, 0, 5, 6, 0, 16, 0, 0, 15, 0, 13, 14, 0, 0, 9, 10, 0, 12, 0, 0, 11, 32, 0, 0, 31, 0, 29, 30, 0, 0, 25, 26, 0, 28, 0, 0, 27, 0, 17, 18, 0, 20, 0, 0, 19, 24, 0, 0, 23, 0, 21, 22, 0, 64, 0, 0, 63, 0, 61, 62, 0, 0, 57, 58, 0, 60, 0, 0, 59, 0, 49, 50, 0, 52

Offset: 0

Antti Karttunen, Aug 18 2014

nonn

Sequence has nonzero values a(n) = k at those points n for which A065621(k) = n and zeros at those positions n which are not present in A065621.

Equally, sequence is obtained when the negative terms of A065620 are replaced with zeros

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A246159, A006068, A010060, A115384, A065620, A065621.

a065620(n) = if(n<3, n, if(n%2, -2*a065620((n - 1)/2) + 1, 2*a065620(n/2)));
a(n) = (hammingweight(n)%2)*a065620(n);
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Jun 07 2017

def a065620(n): return n if n<3 else 2*a065620(n//2) if n%2==0 else -2*a065620((n - 1)//2) + 1
def a(n): return (bin(n)[2:].count("1")%2)*a065620(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(n) = A010060(n) * A065620(n).
a(n) = A246159(n) + A065620(n).
a(0) = 0, and for n >= 1, a(n) = A010060(n) *  (1 + A006068(A115384(n)-1)).
For all n, a(A065621(n)) = n.

A277811 Column 1 of A277810: a(n) = A019565(A065621(n)).

Original entry on oeis.org

2, 3, 30, 5, 70, 105, 42, 7, 154, 231, 2310, 385, 110, 165, 66, 11, 286, 429, 4290, 715, 10010, 15015, 6006, 1001, 182, 273, 2730, 455, 130, 195, 78, 13, 442, 663, 6630, 1105, 15470, 23205, 9282, 1547, 34034, 51051, 510510, 85085, 24310, 36465, 14586, 2431, 374, 561, 5610, 935, 13090, 19635, 7854, 1309, 238, 357, 3570, 595, 170, 255, 102, 17

Offset: 1

Antti Karttunen, Nov 01 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192

Column 1 of A277810.
Permutation of A030059.
Cf. A019565, A065621.

(define (A277811 n) (A019565 (A065621 n)))

a(n) = A019565(A065621(n)).

cp's OEIS Frontend

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A379121 Odd squares k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

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A114390 a(n) = A065621(n^2).

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A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

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A284270 Square array A(r,c) = A048720(A065621(r), c) mod r, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A325570 Numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

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A379114 Numbers k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

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A245471 If n is odd, then a(n) = A065621(n+1). If n is even, then a(n) = n/2.

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A246160 Inverse function to the injection A065621.

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