A057300 -id:A057300 - cp's OEIS frontend

A126007 Involution of nonnegative integers: Keep the least significant quaternary digit q0 of n fixed, but swap the positions of digits q1 <-> q2, q3 <-> q4, ..., etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).

0, 1, 2, 3, 16, 17, 18, 19, 32, 33, 34, 35, 48, 49, 50, 51, 4, 5, 6, 7, 20, 21, 22, 23, 36, 37, 38, 39, 52, 53, 54, 55, 8, 9, 10, 11, 24, 25, 26, 27, 40, 41, 42, 43, 56, 57, 58, 59, 12, 13, 14, 15, 28, 29, 30, 31, 44, 45, 46, 47, 60, 61, 62, 63, 256, 257, 258, 259, 272

Offset: 0

Antti Karttunen, Jan 02 2007

Index entries for sequences that are permutations of the natural numbers

Cf. A057300, A126006, A126008.

#include 
uint32_t a(uint32_t n) { return ((n & 0xcccccccc) << 2) | ((n & 0x33333330) >> 2) | (n & 3); } /* Falk Hüffner, Jan 23 2022 */

f(n) = my(d=Vecrev(digits(n, 4))); if (#d % 2, d = concat(d, 0)); fromdigits(Vecrev(vector(#d, i, d[i+(-1)^(i-1)])), 4); \\ A126006
a(n) = (n % 4) + 4*f(n\4); \\ Michel Marcus, Jan 23 2022

a(n) = (n mod 4) + 4*A126006(floor(n/4)).

a(n) = A057300(A126008(n)) = A126008(A057300(n)).

A163234 Inverse permutation to A163233.

Original entry on oeis.org

0, 1, 2, 4, 6, 3, 11, 7, 9, 13, 5, 8, 24, 18, 17, 12, 28, 21, 37, 29, 10, 15, 16, 22, 58, 48, 47, 38, 31, 39, 23, 30, 35, 43, 27, 34, 62, 52, 51, 42, 14, 19, 20, 26, 32, 25, 41, 33, 112, 98, 97, 84, 73, 85, 61, 72, 70, 59, 83, 71, 40, 49, 50, 60, 120, 105, 137, 121, 78

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A163233. a(n) = A163236(A057300(n)). Cf. A163236.

def A(x, y): return (((x + y)**2) + x + 3*y)//2
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a059905(n): return sum([(n>>2*i&1)<Indranil Ghosh, Jun 25 2017

Scheme

(define (A163234 n) (A001477bi (A006068 (A059905 n)) (A006068 (A059906 n)))) (define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

Formula

a(n) = A001477bi(A006068(A059905(n)),A006068(A059906(n))), where A001477bi(x,y) = (((x+y)^2)+x+(3y))/2.

A163236 Inverse permutation to A163235.

Original entry on oeis.org

0, 2, 1, 4, 9, 5, 13, 8, 6, 11, 3, 7, 24, 17, 18, 12, 35, 27, 43, 34, 14, 20, 19, 26, 62, 51, 52, 42, 32, 41, 25, 33, 28, 37, 21, 29, 58, 47, 48, 38, 10, 16, 15, 22, 31, 23, 39, 30, 112, 97, 98, 84, 70, 83, 59, 71, 73, 61, 85, 72, 40, 50, 49, 60, 135, 119, 151, 134, 90

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A163235. a(n) = A163234(A057300(n)). Cf. A163234.

def A(x, y): return (((x + y)**2) + x + 3*y)//2
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a059905(n): return sum([(n>>2*i&1)<Indranil Ghosh, Jun 25 2017

Scheme

(define (A163236 n) (A001477bi (A006068 (A059906 n)) (A006068 (A059905 n)))) (define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

Formula

a(n) = A001477bi(A006068(A059906(n)),A006068(A059905(n))), where A001477bi(x,y) = (((x+y)^2)+x+(3y))/2.

A302782 Inverse permutation to A302781.

Original entry on oeis.org

0, 1, 3, 15, 5, 2, 21, 14, 63, 6, 255, 12, 85, 20, 4, 341, 1023, 62, 4095, 10, 22, 254, 1365, 13, 5461, 86, 60, 16, 16383, 7, 65535, 340, 252, 1022, 26, 48, 21845, 4094, 84, 9, 87381, 23, 262143, 240, 58, 1366, 1048575, 342, 349525, 5460, 1020, 90, 1398101, 61, 250, 19, 4092, 16382, 4194303, 11, 16777215, 65534, 42

Offset: 1

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Cf. A302781 (inverse).

Cf. A006068, A052331, A163355, A302845.

up_to = 8192;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r))));
A302845(n) = A163355(A006068(A006068(n)));
A302782(n) = A302845(A052331(n));

a(n) = A302845(A052331(n)).

A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 3, 3, 4, 0, 0, 5, 8, 0, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 9, 15, 1, 15, 9, 7, 0, 0, 8, 11, 15, 5, 5, 15, 11, 8, 0, 0, 9, 4, 12, 9, 0, 9, 12, 4, 9, 0, 0, 10, 6, 12, 13, 15, 15, 13, 12, 6, 10, 0

Offset: 0

Rémy Sigrist, Feb 08 2021

For any x >= 0, the function n -> T(n, 2^x) is a self-inverse permutation of the nonnegative integers.
The set of nonnegative integers equipped with T forms a commutative monoid; its invertible elements are the odious numbers (A000069).
Hence A000069 equipped with T forms a group.

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+---------------------------------------------------------------
    0|  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
    1|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    2|  0   2   1   3   8  10   9  11   4   6   5   7  12  14  13  15 -> A057300
    3|  0   3   3   0  12  15  15  12  12  15  15  12   0   3   3   0
    4|  0   4   8  12   1   5   9  13   2   6  10  14   3   7  11  15 -> A126006
    5|  0   5  10  15   5   0  15  10  10  15   0   5  15  10   5   0
    6|  0   6   9  15   9  15   0   6   6   0  15   9  15   9   6   0
    7|  0   7  11  12  13  10   6   1  14   9   5   2   3   4   8  15
    8|  0   8   4  12   2  10   6  14   1   9   5  13   3  11   7  15
    9|  0   9   6  15   6  15   0   9   9   0  15   6  15   6   9   0
   10|  0  10   5  15  10   0  15   5   5  15   0  10  15   5  10   0
   11|  0  11   7  12  14   5   9   2  13   6  10   1   3   8   4  15
   12|  0  12  12   0   3  15  15   3   3  15  15   3   0  12  12   0
   13|  0  13  14   3   7  10   9   4  11   6   5   8  12   1   2  15
   14|  0  14  13   3  11   5   6   8   7   9  10   4  12   2   1  15
   15|  0  15  15   0  15   0   0  15  15   0   0  15   0  15  15   0
                                                                     \
                                                                      v
                                                                    A010060

Rémy Sigrist, Table of n, a(n) for n = 0..10010 (rows 0..140)
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Rémy Sigrist, Colored representation of the table over the first 128 odious numbers
Rémy Sigrist, Colored representation of the table over the first 1024 evil numbers (white pixels correspond to 0's)

Cf. A000069, A010060, A057300, A126006, A133457.

B(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n -= 2^(b[k] = valuation(n, 2))); b }
T(n,k) = { my (nn=B(n), kk=B(k), v=0); for (i=1, #nn, for (j=1, #kk, v=bitxor(v, 2^bitxor(nn[i], kk[j])))); v }

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, 0) = 0 (0 is an absorbing element for T).
T(n, 1) = n (1 is the neutral element for T).
T(n, 2) = A057300(n).
T(n, 4) = A126006(n).
T(n, n) = A010060(n).
A010060(T(n, k)) = A010060(n) * A010060(k).

A341487 Second row of A341458.

Original entry on oeis.org

2, 1, 5, 6, 3, 4, 8, 7, 17, 18, 22, 21, 20, 19, 23, 24, 9, 10, 14, 13, 12, 11, 15, 16, 26, 25, 29, 30, 27, 28, 32, 31, 65, 66, 70, 69, 68, 67, 71, 72, 82, 81, 85, 86, 83, 84, 88, 87, 74, 73, 77, 78, 75, 76, 80, 79, 89, 90, 94, 93, 92, 91, 95, 96, 33, 34, 38

Offset: 1

Rémy Sigrist, Feb 13 2021

This sequence is a self-inverse permutation of the positive integers.

This sequence has similarities with A057300.

			a(7) = A341458(2, 7) = 8.

Rémy Sigrist, Table of n, a(n) for n = 1..2048
Rémy Sigrist, PARI program for A341487
Index entries for sequences that are permutations of the natural numbers

Cf. A057300, A341458.

PARI
```
See Links section.
```

a(n) = A341458(2, n).

A361946 If the base-4 expansion of n starts with the digit 1, then replace 2's by 3's and vice versa; if it starts with the digit 2, then replace 1's by 3's and vice versa; if it starts with the digit 3, then replace 1's by 2's and vice versa; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 11, 10, 9, 12, 14, 13, 15, 16, 17, 19, 18, 20, 21, 23, 22, 28, 29, 31, 30, 24, 25, 27, 26, 32, 35, 34, 33, 44, 47, 46, 45, 40, 43, 42, 41, 36, 39, 38, 37, 48, 50, 49, 51, 56, 58, 57, 59, 52, 54, 53, 55, 60, 62, 61, 63, 64, 65, 67, 66

Offset: 0

Rémy Sigrist, Apr 01 2023

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 539:
- the base-4 expansion of 539 is "20123",
- it starts with the digit 2, so we replace 1's by 3's and vice versa,
- so the base-4 expansion of a(539) is "20321", and a(539) = 569.

Cf. A000695, A048647, A057300, A122587, A163241, A361945, A361947.

a(n) = { my (q = digits(n, 4), m = if (#q, [ [0,1,3,2], [0,3,2,1], [0,2,1,3] ][q[1]], [0,1,2,3])); fromdigits(apply (d -> m[1+d], q), 4); }

a(n) = A163241(n) when A122587(n) = 1.
a(n) = A048647(n) when A122587(n) = 2.
a(n) = A057300(n) when A122587(n) = 3.
a(n) = n iff n = d * A000695(k) for some d in {1, 2, 3} and some k >= 0.

A361947 If the rightmost nonzero digit in the base-4 expansion of n is the digit 1, then replace 2's by 3's and vice versa; if it is the digit 2, then replace 1's by 3's and vice versa; if it is the digit 3, then replace 1's by 2's and vice versa; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 14, 11, 8, 13, 10, 7, 12, 9, 6, 15, 16, 17, 50, 35, 20, 21, 62, 43, 56, 29, 58, 39, 44, 25, 54, 47, 32, 49, 34, 19, 52, 53, 46, 27, 40, 61, 42, 23, 28, 57, 38, 31, 48, 33, 18, 51, 36, 37, 30, 59, 24, 45, 26, 55, 60, 41, 22, 63, 64, 65, 194

Offset: 0

Rémy Sigrist, Apr 01 2023

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 539:
- the base-4 expansion of 539 is "20123",
- the rightmost nonzero digit is 3, so we replace 1's by 2's and vice versa,
- so the base-4 expansion of a(539) is "10213", and a(539) = 295.

Cf. A000695, A048647, A057300, A065882, A163241, A361945, A361946.

a(n) = { my (m = if (n, [ [0,1,3,2], [0,3,2,1], [0,2,1,3] ][(n / 4^valuation(n, 4)) % 4], [0,1,2,3])); fromdigits(apply (d -> m[1+d], digits(n, 4)), 4); }

a(n) = A163241(n) when A065882(n) = 1.
a(n) = A048647(n) when A065882(n) = 2.
a(n) = A057300(n) when A065882(n) = 3.
a(n) = n iff n = d * A000695(k) for some d in {1, 2, 3} and some k >= 0.

A057301 Binary counter with bit positions permuted in increasing length cycles: 0;2,1;5,3,4;9,6,7,8,...

Original entry on oeis.org

0, 1, 4, 5, 2, 3, 6, 7, 32, 33, 36, 37, 34, 35, 38, 39, 8, 9, 12, 13, 10, 11, 14, 15, 40, 41, 44, 45, 42, 43, 46, 47, 16, 17, 20, 21, 18, 19, 22, 23, 48, 49, 52, 53, 50, 51, 54, 55, 24, 25, 28, 29, 26, 27, 30, 31, 56, 57, 60, 61, 58, 59, 62, 63, 512, 513, 516, 517, 514, 515

Offset: 0

Marc LeBrun, Aug 24 2000

A permutation of the integers

A057300.

cp's OEIS Frontend

A126007 Involution of nonnegative integers: Keep the least significant quaternary digit q0 of n fixed, but swap the positions of digits q1 <-> q2, q3 <-> q4, ..., etc. in the base-4 expansion of n (where n = ... + q4*256 + q3*64 + q2*16 + q1*4 + q0).

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A163234 Inverse permutation to A163233.

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A163236 Inverse permutation to A163235.

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A302782 Inverse permutation to A302781.

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A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

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A341487 Second row of A341458.

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A361946 If the base-4 expansion of n starts with the digit 1, then replace 2's by 3's and vice versa; if it starts with the digit 2, then replace 1's by 3's and vice versa; if it starts with the digit 3, then replace 1's by 2's and vice versa; a(0) = 0.

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A361947 If the rightmost nonzero digit in the base-4 expansion of n is the digit 1, then replace 2's by 3's and vice versa; if it is the digit 2, then replace 1's by 3's and vice versa; if it is the digit 3, then replace 1's by 2's and vice versa; a(0) = 0.

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A126007 Involution of nonnegative integers: Keep the least significant quaternary digit q0 of n fixed, but swap the positions of digits q1 <-> q2, q3 <-> q4, ..., etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).