A233279 -id:A233279 - cp's OEIS frontend

A064707 Inverse square of permutation defined by A003188.

0, 1, 2, 3, 5, 4, 7, 6, 10, 11, 8, 9, 15, 14, 13, 12, 21, 20, 23, 22, 16, 17, 18, 19, 31, 30, 29, 28, 26, 27, 24, 25, 42, 43, 40, 41, 47, 46, 45, 44, 32, 33, 34, 35, 37, 36, 39, 38, 63, 62, 61, 60, 58, 59, 56, 57, 53, 52, 55, 54, 48, 49, 50, 51, 85, 84, 87, 86, 80, 81, 82, 83

Offset: 0

N. J. A. Sloane, Oct 13 2001

Not the same as A100281: a(n)=A100281(n)=A099896(A099896(n)) only for n<64. - Reinhard Zumkeller, Nov 11 2004

Inverse of permutation defined by A064706. Cf. A003188.

A = 1; for i = 1:7 B = A(end:-1:1); A = [A (B + length(A))]; end C = A(A); for i = 1:128 A(C(i)) = i - 1; end A

a(n) = A180200(A233279(n)), n > 0. - Yosu Yurramendi, Apr 05 2017

More terms from David Wasserman, Aug 02 2002

A246162 Permutation of natural numbers: a(1) = 1, a(A000069(n)) = A014580(a(n-1)), a(A001969(n)) = A091242(a(n-1)), where A000069 and A001969 are the odious and evil numbers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 8, 11, 7, 6, 9, 13, 14, 31, 47, 17, 25, 12, 10, 19, 15, 37, 59, 20, 21, 61, 185, 42, 319, 62, 24, 87, 137, 34, 18, 55, 16, 41, 97, 27, 22, 67, 229, 49, 415, 76, 28, 103, 29, 109, 425, 78, 1627, 222, 54, 283, 3053, 373, 79, 433, 33, 131, 647, 108, 1123, 166, 45, 203, 26, 91, 379, 71, 23

Offset: 1

Antti Karttunen, Aug 17 2014. Erroneous comment corrected Aug 20 2014

nonn

This is an instance of entanglement-permutation, where the two complementary pairs to be entangled with each other are A000069/A001969 (odious and evil numbers) and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).

Because 3 is the only evil number in A014580, it implies that, apart from a(4)=3, all other odious positions contain an odious number. There are also odious numbers in some of the evil positions, precisely all the terms of A246158 in some order, together with all evil numbers larger than 3. (Permutation A246164 has the same property, except there a(7)=3.) See comments in A246161 for more details how this affects the cycle structure of these permutations.

Inverse: A246161.
Related permutations: A006068, A233279, A234612, A237427, A245702, A246202, A246164.
Cf. A010060, A014580, A091242, A115384, A245710.

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. n is one of the odious numbers, A000069], a(n) = A014580(a(A115384(n)-1)), otherwise, a(n) = A091242(a(A245710(n))).
As a composition of related permutations:
a(n) = A245702(A233279(n)).
a(n) = A246202(A006068(n)).
a(n) = A246164(A234612(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].

A324337 a(n) = A002487(A006068(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 3, 4, 3, 2, 5, 1, 4, 5, 3, 5, 4, 3, 7, 2, 7, 8, 5, 1, 5, 7, 4, 7, 5, 3, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13, 17, 10, 15, 11, 7, 18, 2, 11, 16, 9, 17, 12, 7, 19, 14, 11, 8, 19, 5, 18, 21, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324338, a few terms preceding each position n = 2^k seem to be a batch of nearby Fibonacci numbers in some order.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A002487, A006068, A063946, A071585, A248663, A283477, A324287, A324338.

Block[{f}, f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a + b, a = a + b]; n = Floor[n/2]]; b]; Array[f@ Fold[BitXor, #, Quotient[#, 2^Range[BitLength[#] - 1]]] &, 106, 0]] (* Michael De Vlieger, Dec 14 2019, after Jean-François Alcover at A002487 and Jan Mangaldan at A006068 *)

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
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324337(n) = A002487(A006068(n));

maxlevel <- 6 # by choice
#
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+        k) = A324338(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1)
a(2^m+2^(m-1)+k) = A324338(2^m+        k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324338(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A248663(A283477(n))). - Antti Karttunen, Nov 06 2019
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324338(k), m >= 0, 0 <= k < 2^m.
a(A059893(2^(m+1)+A001969(k+1))) - a(A059893(2^m+A001969(k+1))) = A071585(k), m >= 0, 0 <= k < 2^(m-1).
a(A059893(2^(m+1)+ A000069(k+1))) = A071585(k), m >= 1, 0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Dec 27 2019

A324338 a(n) = A002487(1+A006068(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 2, 1, 4, 5, 3, 4, 3, 2, 5, 1, 5, 7, 4, 7, 5, 3, 8, 5, 4, 3, 7, 2, 7, 8, 5, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10, 7, 17, 4, 15, 18, 11, 11, 9, 7, 16, 5, 17, 19, 12, 3, 14, 19, 11, 18, 13, 8, 21, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324337, a few terms preceding each 2^k-th term (here always 1) seem to consist of a batch of nearby Fibonacci numbers (A000045) in some order. For example, a(65533) = 987, a(65534) = 610 and a(65535) = 1597.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A000045, A002487, A006068, A324288, A324337.

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
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324338(n) = A002487(1+A006068(n));

maxlevel <- 6 # by choice #
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

a(n) = A002487(1+A006068(n)).
a(2^n) = 1 for all n >= 0.
From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+2^(m-1)+k) = A324337(2^m+        k), m > 0, 0 <= k < 2^(m-1)
a(2^m+        k) = A324337(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324337(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A233279(n)), n > 0. Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324337(k), m >= 0,  0 <= k < 2^m.
a(A059893(2^(m+1)+A000069(k+1))) - a(A059893(2^m+A000069(k+1))) =  A071585(k), m >= 1,  0 <= k < 2^(m-1).
a(A059893(2^m+ A001969(k+1))) = A071585(k),    m >= 0,  0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(A233279(n)). Yosu Yurramendi, Dec 27 2019

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A064707 Inverse square of permutation defined by A003188.

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A246162 Permutation of natural numbers: a(1) = 1, a(A000069(n)) = A014580(a(n-1)), a(A001969(n)) = A091242(a(n-1)), where A000069 and A001969 are the odious and evil numbers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

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A324337 a(n) = A002487(A006068(n)).

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A324338 a(n) = A002487(1+A006068(n)).

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