A064707 -id:A064707 - cp's OEIS frontend

A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

0, 1, 3, 2, 15, 14, 12, 13, 5, 6, 4, 7, 10, 9, 11, 8, 21, 20, 22, 23, 16, 19, 17, 18, 26, 27, 25, 24, 31, 28, 30, 29, 63, 62, 60, 61, 48, 49, 51, 50, 58, 57, 59, 56, 53, 54, 52, 55, 42, 43, 41, 40, 47, 44, 46, 45, 37, 36, 38, 39, 32, 35, 33, 34, 255, 254, 252, 253, 240, 241, 243, 242, 250, 249, 251, 248, 245, 246, 244

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302846 (inverse).

Cf. A006068, A057300, A064707, A163355, A302843, A302782.

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
A064707(n) = A006068(A006068(n));
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(A064707(n));

a(n) = A163355(A064707(n)).

a(n) = A302843(A006068(n)).

A064706 Square of permutation defined by A003188.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 13 2001

Inverse of sequence A064707 considered as a permutation of the nonnegative integers.

Not the same as A100282: a(n) = A100282(n) = A100280(A100280(n)) only for n < 64. - Reinhard Zumkeller, Nov 11 2004

Cf. A064707 (inverse), A165211 (mod 2).
Cf. A003188, A000695, A059905, A059906.
Cf. also A054238, A163233, A302846.

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

Array[BitXor[#, Floor[#/4]] &, 72, 0] (* Michael De Vlieger, Apr 14 2018 *)

PARI
```
a(n)=bitxor(n,n\4)
```

{ for (n=0, 1000, write("b064706.txt", n, " ", bitxor(n, n\4)) ) } \\ Harry J. Smith, Sep 22 2009

def A064706(n): return n^ n>>2 # Chai Wah Wu, Jun 29 2022

maxn <- 63 # by choice
b <- c(1,0,0)
for(n in 4:maxn) b[n] <- b[n-1] - b[n-2] + b[n-3]
# c(1,b) is A133872
a <- 1
for(n in 1:maxn) {
a[2*n  ] <- 2*a[n] + 1 - b[n]
a[2*n+1] <- 2*a[n] +     b[n]
}
(a <- c(0,a))
# Yosu Yurramendi, Oct 25 2020

a(n) = A003188(A003188(n)).
a(n) = n XOR floor(n/4), where XOR is binary exclusive OR. - Paul D. Hanna, Oct 25 2004
a(n) = A233280(A180201(n)), n > 0. - Yosu Yurramendi, Apr 05 2017
a(n) = A000695(A003188(A059905(n))) + 2*A000695(A003188(A059906(n))). - Antti Karttunen, Apr 14 2018

More terms from David Wasserman, Aug 02 2002

A302783 A divisor-or-multiple permutation of natural numbers: a(n) = A052330(A006068(n)).

Original entry on oeis.org

1, 2, 6, 3, 24, 12, 4, 8, 120, 60, 20, 40, 5, 10, 30, 15, 840, 420, 140, 280, 35, 70, 210, 105, 7, 14, 42, 21, 168, 84, 28, 56, 7560, 3780, 1260, 2520, 315, 630, 1890, 945, 63, 126, 378, 189, 1512, 756, 252, 504, 9, 18, 54, 27, 216, 108, 36, 72, 1080, 540, 180, 360, 45, 90, 270, 135, 83160, 41580, 13860, 27720, 3465, 6930, 20790, 10395, 693

Offset: 0

Antti Karttunen, Apr 16 2018

nonn

Shares with A064736, A207901, A302781, A302350, etc. a property that a(n) is always either a divisor or a multiple of a(n+1). However, because multiple bits may change simultaneously when moving from A006068(n) to A006068(n+1) [with the restriction that the changing bits are all either toggled on or all toggled off], it means that also here the terms might be divided or multiplied by more than just a single Fermi-Dirac prime (A050376). E.g. a(3) = 3, while a(4) = A050376(1) * A050376(3) * 3 = 2*4*3 = 24. See also comments in A284003.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Cf. A302784 (inverse).
Cf. A006068, A050376, A052330, A302777.
Cf. also A207901 and A284003 (a squarefree analog).

up_to_e = 13;
v050376 = vector(up_to_e);
A050376(n) = v050376[n];
A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
i = 0; for(n=1,oo,if(A302777(n), i++; v050376[i] = n); if(i == up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
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
A302783(n) = A052330(A006068(n));

a(n) = A052330(A006068(n)).

a(n) = A207901(A064707(n)).

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 15, 5, 10, 210, 105, 35, 70, 7, 14, 42, 21, 2310, 1155, 385, 770, 77, 154, 462, 231, 11, 22, 66, 33, 330, 165, 55, 110, 30030, 15015, 5005, 10010, 1001, 2002, 6006, 3003, 143, 286, 858, 429, 4290, 2145, 715, 1430, 13, 26, 78, 39, 390, 195, 65, 130, 2730, 1365, 455, 910, 91, 182, 546, 273, 510510, 255255, 85085, 170170, 17017

Offset: 0

Antti Karttunen, Mar 18 2017

A squarefree analog of A302783. Each term is either a divisor or a multiple of the next one. In contrast to A302033 at each step the previous term can be multiplied (or divided), not just by a single prime, but possibly by a product of several distinct ones, A019565(A000975(k)). E.g., a(3) = 3, a(4) = 2*5*a(3) = 30. - Antti Karttunen, Apr 17 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

A permutation of A005117.
Cf. A000975, A001222, A006068, A007913, A019565, A046523, A048675, A064707, A209281, A283477, A284004.
Cf. also A277811, A283475, A302033, A302783.

Table[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Times @@ (p^Mod[e, 2])] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 52}] (* Michael De Vlieger, Mar 18 2017 *)

A007913(n) = core(n);
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A283477(n) = A108951(A019565(n));
A284003(n) = A007913(A283477(n));

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
A284003(n) = A019565(A006068(n)); \\ (and use A019565 from above) - Antti Karttunen, Apr 16 2018

(define (A284003 n) (A007913 (A283477 n)))

a(n) = A007913(A283477(n)).
Other identities. For all n >= 0:
A048675(a(n)) = A006068(n).
A046523(a(n)) = A284004(n).
It seems that A001222(a(n)) = A209281(n).
a(n) = A019565(A006068(n)) = A302033(A064707(n)). - Antti Karttunen, Apr 16 2018

Name amended with a second formula by Antti Karttunen, Apr 16 2018

A100281 a(n) = A099896(A099896(n)).

Original entry on oeis.org

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, 84, 85, 86, 87, 81, 80, 83, 82, 94

Offset: 0

Reinhard Zumkeller, Nov 11 2004

Permutation of the natural numbers with inverse A100282;

A064707(n) = a(n) for n<64.

Cf. A099896, A064707, A100282.

def A100281(n): return n^(m:=n>>2)^(m>>2) # Chai Wah Wu, Jan 19 2023

a(n) = n XOR floor(n/4) XOR floor(n/16). - Ivan Neretin, Sep 06 2017

cp's OEIS Frontend

A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

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A064706 Square of permutation defined by A003188.

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A302783 A divisor-or-multiple permutation of natural numbers: a(n) = A052330(A006068(n)).

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A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

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A100281 a(n) = A099896(A099896(n)).

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