Displaying 1-10 of 152 results found.
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0, 1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14, 48, 16, 19, 17, 21, 23, 18, 22, 25, 27, 30, 26, 20, 28, 31, 29, 96, 32, 35, 33, 37, 39, 34, 38, 41, 43, 46, 42, 36, 44, 47, 45, 49, 51, 54, 50, 60, 52, 55, 53, 40, 56, 59, 57, 61, 63, 58, 62, 192, 64, 67, 65, 69, 71, 66, 70, 73, 75, 78, 74, 68, 76, 79, 77, 81
FORMULA
Other identities. For all n >= 0:
A101080(n,a(n+1)) = 1. [The Hamming distance between n and a(n+1) is always one.]
MATHEMATICA
A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := If[n == 0, 0, BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]]; a[n_] := If[n == 0, 0, A003188[1 + A006068[n-1]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)
PROG
(PARI) A003188(n) = bitxor(n, floor(n/2));
A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2});
(Python)
if n<2: return n
return 2*m + (n%2 + m%2)%2
(Python)
k, m = n-1, n-1>>1
while m > 0:
k ^= m
m >>= 1
CROSSREFS
Row 1 and column 1 of array A268715 (without the initial zero).
Cf. A268817 ("square" of this permutation).
0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 10, 15, 14, 12, 13, 16, 17, 19, 18, 23, 22, 20, 21, 31, 30, 28, 29, 24, 25, 27, 26, 32, 33, 35, 34, 39, 38, 36, 37, 47, 46, 44, 45, 40, 41, 43, 42, 63, 62, 60, 61, 56, 57, 59, 58, 48, 49, 51, 50, 55, 54, 52, 53, 64, 65, 67, 66
MATHEMATICA
Module[{nn = 6, s}, s = Flatten[Table[Range[2^(n + 1) - 1, 2^n, -1], {n, 0, nn}]]; Map[If[# == 0, 0, s[[#]]] &, Table[Fold[BitXor, n, Quotient[n, 2^Range[BitLength[n] - 1]]], {n, 0, 2^nn}]]] (* Michael De Vlieger, Apr 06 2017, after Harvey P. Dale at A054429 and Jan Mangaldan at A006068 *)
PROG
(R)
maxrow <- 8 # by choice
a <- 1:3
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+2)+ k] <- a[2^(m+1)+ k] + 2^(m+1)
a[2^(m+2)+ 2^m+k] <- a[2^(m+1)+2^m+k] + 2^(m+1)
a[2^(m+2)+2^(m+1)+ k] <- a[2^(m+1)+2^m+k] + 2^(m+2)
a[2^(m+2)+2^(m+1)+2^m+k] <- a[2^(m+1)+ +k] + 2^(m+2)
}
(a <- c(0, a))
(R)
# Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 7 # by choice
a <- 1
for(n in 2:2^maxblock){
ones <- which(as.integer(intToBits(n)) == 1)
nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
anbit <- nbit
for(k in 2^(0:floor(log2(length(nbit)))) )
anbit <- bitwXor(anbit, c(anbit[-(1:k)], rep(0, k))) # ?bitwXor
anbit[0:(length(anbit) - 1)] <- 1 - anbit[0:(length(anbit)-1)]
a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
(a <- c(0, a))
(Python)
from sympy import floor
def a006068(n):
s=1
while True:
ns=n>>s
if ns==0: break
n=n^ns
s<<=1
return n
def a054429(n): return 1 if n==1 else 2*a054429(floor(n/2)) + 1 - n%2
def a(n): return 0 if n==0 else a054429( a006068(n)) # Indranil Ghosh, Jun 11 2017
0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11, 5, 13, 16, 14, 18, 20, 23, 19, 29, 21, 24, 22, 9, 25, 28, 26, 30, 32, 27, 31, 34, 36, 39, 35, 45, 37, 40, 38, 57, 41, 44, 42, 46, 48, 43, 47, 17, 49, 52, 50, 54, 56, 51, 55, 58, 60, 63, 59, 53, 61, 64, 62, 66, 68, 71, 67, 77, 69, 72, 70, 89, 73, 76, 74, 78, 80, 75, 79, 113, 81
FORMULA
Other identities. For all n >= 1:
a( A128309(n)) = A128309(n)+2. [Maps any even odious number to that number + 2.]
PROG
(PARI)
a003188(n)=bitxor(n, n>>1);
a006068(n)= {
my( s=1, ns );
while ( 1,
ns = n >> s;
if ( 0==ns, break() );
n = bitxor(n, ns);
s <<= 1;
);
return (n);
a(n)=if(n==0, 0, 1 + a003188( a006068(n) - 1)); \\ Indranil Ghosh, Jun 07 2017
(Python)
def a003188(n): return n^(n>>1)
def a006068(n):
s=1
while True:
ns=n>>s
if ns==0: break
n=n^ns
s<<=1
return n
def a(n): return 0 if n==0 else 1 + a003188( a006068(n) - 1) # Indranil Ghosh, Jun 07 2017
CROSSREFS
Cf. A268818 ("square" of this permutation).
Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = A003188(1+ A006068(A(r-1,c-1))) = A268717(1+A(r-1,c-1)), read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...
+20
13
0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 3, 1, 0, 5, 2, 2, 3, 1, 0, 6, 12, 7, 2, 3, 1, 0, 7, 4, 6, 6, 2, 3, 1, 0, 8, 7, 13, 5, 6, 2, 3, 1, 0, 9, 5, 12, 7, 7, 6, 2, 3, 1, 0, 10, 24, 5, 15, 4, 7, 6, 2, 3, 1, 0, 11, 8, 4, 13, 5, 5, 7, 6, 2, 3, 1, 0, 12, 11, 25, 4, 14, 12, 5, 7, 6, 2, 3, 1, 0, 13, 9, 24, 12, 15, 4, 4, 5, 7, 6, 2, 3, 1, 0, 14, 13, 9, 27, 12, 10, 13, 4, 5, 7, 6, 2, 3, 1, 0
FORMULA
For row zero: A(0,k) = k, for column zero: A(n,0) = 0, and in other cases: A(n,k) = A003188(1+ A006068(A(n-1,k-1)))
Other identities. For all n >= 0:
EXAMPLE
The top left [0 .. 16] x [0 .. 19] section of the array:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
0, 1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14, 48, 16, 19
0, 1, 3, 2, 7, 6, 13, 12, 5, 4, 25, 24, 9, 8, 15, 14, 11, 10, 49, 48
0, 1, 3, 2, 6, 5, 7, 15, 13, 4, 12, 27, 25, 8, 24, 14, 10, 9, 11, 51
0, 1, 3, 2, 6, 7, 4, 5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11, 8, 9
0, 1, 3, 2, 6, 7, 5, 12, 4, 10, 14, 13, 15, 30, 26, 25, 27, 11, 9, 24
0, 1, 3, 2, 6, 7, 5, 4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26, 9, 8
0, 1, 3, 2, 6, 7, 5, 4, 12, 15, 13, 9, 11, 14, 10, 29, 31, 26, 30, 8
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 14, 15, 8, 9, 10, 11, 28, 29, 30, 31
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 10, 14, 24, 8, 11, 9, 20, 28, 31
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 11, 10, 25, 24, 9, 8, 21, 20
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 9, 11, 27, 25, 8, 24, 23
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 8, 9, 26, 27, 24, 25
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 24, 8, 30, 26, 25
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 25, 24, 31, 30
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 24, 27, 25, 29
0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 24, 25, 26, 27
MATHEMATICA
A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m= A006068[Floor[n/2]]}, 2m + Mod[Mod[n, 2] + Mod[m, 2], 2]]]; a[r_, 0]:= 0; a[0, c_]:=c; a[r_, c_]:= A003188[1 + A006068[a[r - 1, c - 1]]]; Table[a[c, r - c], {r, 0, 15}, {c, 0, r}] //Flatten (* Indranil Ghosh, Apr 02 2017 *)
PROG
(Scheme)
(define (A268820bi row col) (cond ((zero? row) col) ((zero? col) 0) (else ( A268717 (+ 1 (A268820bi (- row 1) (- col 1)))))))
(define (A268820bi row col) (cond ((zero? row) col) ((zero? col) 0) (else ( A003188 (+ 1 ( A006068 (A268820bi (- row 1) (- col 1))))))))
(PARI) A003188(n) = bitxor(n, n\2);
a(r, c) = if(r==0, c, if(c==0, 0, A003188(1 + A006068(a(r - 1, c - 1)))));
for(r=0, 15, for(c=0, r, print1(a(c, r - c), ", "); ); print(); ); \\ Indranil Ghosh, Apr 02 2017
(Python)
if n<2: return n
else:
return 2*m + (n%2 + m%2)%2
def a(r, c): return c if r<1 else 0 if c<1 else A003188(1 + A006068(a(r - 1, c - 1)))
for r in range(16):
print([a(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 02 2017
CROSSREFS
Inverses of these permutations can be found in table A268830.
Rows converge towards A003188, which is also the main diagonal.
Cf. array A268715 (can be extracted from this one).
Cf. array A268833 (shows related Hamming distances with regular patterns).
Square array A(i,j) = A003188( A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...
+20
11
0, 1, 1, 2, 3, 2, 3, 6, 6, 3, 4, 2, 5, 2, 4, 5, 12, 7, 7, 12, 5, 6, 4, 15, 6, 15, 4, 6, 7, 7, 13, 13, 13, 13, 7, 7, 8, 5, 4, 12, 9, 12, 4, 5, 8, 9, 24, 12, 5, 11, 11, 5, 12, 24, 9, 10, 8, 27, 4, 14, 10, 14, 4, 27, 8, 10, 11, 11, 25, 25, 10, 15, 15, 10, 25, 25, 11, 11, 12, 9, 8, 24, 29, 14, 12, 14, 29, 24, 8, 9, 12, 13, 13, 24, 9, 31, 31, 13, 13, 31, 31, 9, 24, 13, 13
EXAMPLE
The top left [0 .. 15] x [0 .. 15] section of the array:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14
2, 6, 5, 7, 15, 13, 4, 12, 27, 25, 8, 24, 14, 10, 9, 11
3, 2, 7, 6, 13, 12, 5, 4, 25, 24, 9, 8, 15, 14, 11, 10
4, 12, 15, 13, 9, 11, 14, 10, 29, 31, 26, 30, 8, 24, 27, 25
5, 4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26, 9, 8, 25, 24
6, 7, 4, 5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11, 8, 9
7, 5, 12, 4, 10, 14, 13, 15, 30, 26, 25, 27, 11, 9, 24, 8
8, 24, 27, 25, 29, 31, 26, 30, 17, 19, 22, 18, 28, 20, 23, 21
9, 8, 25, 24, 31, 30, 27, 26, 19, 18, 23, 22, 29, 28, 21, 20
10, 11, 8, 9, 26, 27, 24, 25, 22, 23, 20, 21, 30, 31, 28, 29
11, 9, 24, 8, 30, 26, 25, 27, 18, 22, 21, 23, 31, 29, 20, 28
12, 13, 14, 15, 8, 9, 10, 11, 28, 29, 30, 31, 24, 25, 26, 27
13, 15, 10, 14, 24, 8, 11, 9, 20, 28, 31, 29, 25, 27, 30, 26
14, 10, 9, 11, 27, 25, 8, 24, 23, 21, 28, 20, 26, 30, 29, 31
15, 14, 11, 10, 25, 24, 9, 8, 21, 20, 29, 28, 27, 26, 31, 30
MATHEMATICA
A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := BitXor @@ Table[Floor[ n/2^m], {m, 0, Log[2, n]}]; A006068[0]=0; A[i_, j_] := A003188[ A006068[i] + A006068[j]]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)
PROG
(Scheme)
;; Alternatively, extracting data from array A268820:
(define (A268715bi row col) (A268820bi ( A006068 row) (+ ( A006068 row) col)))
(Python)
def a003188(n): return n^(n>>1)
def a006068(n):
s=1
while True:
ns=n>>s
if ns==0: break
n=n^ns
s<<=1
return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017
CROSSREFS
Cf. A268719 (the lower triangular section).
A divisor-or-multiple permutation of natural numbers: a(n) = A052330( A006068(n)).
+20
10
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
COMMENTS
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.
PROG
(PARI)
up_to_e = 13;
v050376 = vector(up_to_e);
A209229(n) = (n && !bitand(n, n-1));
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
Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1, A268718(c)-1) = 1 + A(r-1, A003188( A006068(c)-1)), read by descending antidiagonals.
+20
9
0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 2, 3, 1, 0, 5, 6, 2, 3, 1, 0, 6, 8, 9, 2, 3, 1, 0, 7, 3, 8, 9, 2, 3, 1, 0, 8, 7, 5, 5, 6, 2, 3, 1, 0, 9, 10, 4, 4, 7, 8, 2, 3, 1, 0, 10, 12, 13, 6, 4, 6, 7, 2, 3, 1, 0, 11, 15, 12, 13, 5, 4, 6, 7, 2, 3, 1, 0, 12, 11, 17, 17, 18, 5, 4, 6, 7, 2, 3, 1, 0, 13, 5, 16, 16, 19, 20, 5, 4, 6, 7, 2, 3, 1, 0, 14, 13, 7, 18, 16, 18, 19, 5, 4, 6, 7, 2, 3, 1, 0
EXAMPLE
The top left [0 .. 16] x [0 .. 19] section of the array:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11, 5, 13, 16, 14, 18, 20, 23, 19
0, 1, 3, 2, 9, 8, 5, 4, 13, 12, 17, 16, 7, 6, 15, 14, 21, 20, 25, 24
0, 1, 3, 2, 9, 5, 4, 6, 13, 17, 16, 18, 10, 8, 15, 7, 21, 25, 24, 26
0, 1, 3, 2, 6, 7, 4, 5, 18, 19, 16, 17, 10, 11, 8, 9, 26, 27, 24, 25
0, 1, 3, 2, 8, 6, 4, 5, 20, 18, 9, 17, 7, 11, 10, 12, 28, 26, 33, 25
0, 1, 3, 2, 7, 6, 4, 5, 19, 18, 11, 10, 9, 8, 13, 12, 27, 26, 35, 34
0, 1, 3, 2, 7, 6, 4, 5, 19, 11, 14, 12, 8, 10, 13, 9, 27, 35, 38, 36
0, 1, 3, 2, 7, 6, 4, 5, 12, 13, 14, 15, 8, 9, 10, 11, 36, 37, 38, 39
0, 1, 3, 2, 7, 6, 4, 5, 14, 16, 11, 15, 8, 9, 12, 10, 38, 40, 35, 39
0, 1, 3, 2, 7, 6, 4, 5, 17, 16, 13, 12, 8, 9, 11, 10, 41, 40, 37, 36
0, 1, 3, 2, 7, 6, 4, 5, 17, 13, 12, 14, 8, 9, 11, 10, 41, 37, 36, 38
0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 12, 13, 8, 9, 11, 10, 38, 39, 36, 37
0, 1, 3, 2, 7, 6, 4, 5, 16, 14, 12, 13, 8, 9, 11, 10, 40, 38, 21, 37
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 39, 38, 23, 22
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 39, 23, 26, 24
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 24, 25, 26, 27
PROG
(Scheme)
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (- ( A268718 col) 1))))))
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) ( A003188 (+ -1 ( A006068 col))))))))
(Python)
def a003188(n): return n^(n>>1)
def a006068(n):
s=1
while True:
ns=n>>s
if ns==0: break
n=n^ns
s<<=1
return n
def a278618(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1)
def A(r, c): return c if r==0 else 0 if c==0 else 1 + A(r - 1, a278618(c) - 1)
for r in range(21): print([A(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Jun 07 2017
CROSSREFS
Inverses of these permutations can be found in table A268820.
Square array A(n, k) = A101080(k, A003188(n+ A006068(k))), read by descending antidiagonals, where A003188 is the binary Gray code, A006068 is its inverse, and A101080(x,y) gives the Hamming distance between binary expansions of x and y.
+20
8
0, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 3, 2, 0, 1, 2, 3, 2, 3, 0, 1, 2, 1, 2, 1, 2, 0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 3, 2, 0, 1, 2, 1, 2, 3, 4, 3, 2, 3, 0, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 0, 1, 2, 3, 2, 3, 4, 3, 2, 1, 4, 3, 0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 0, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 0, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 1, 2, 1, 2
COMMENTS
The entry at row n, column k, gives the Hamming distance between binary expansions of k and A003188(n+ A006068(k)). When Gray code is viewed as a traversal of vertices of an infinite dimensional hypercube by bit-flipping (see the illustration "Visualized as a traversal of vertices of a tesseract" in the Wikipedia's "Gray code" article) the argument k is the "address" (the binary code given inside each vertex) of the starting vertex, and argument n tells how many edges forward along the Gray code path we should hop from it (to the direction that leads away from the vertex with code 0000...). A(n, k) gives then the Hamming distance between the starting and the ending vertex. For how this works with case n=3, see comments in A268676. - Antti Karttunen, Mar 11 2024
EXAMPLE
The top left [0 .. 24] X [0 .. 24] section 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
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 3, 3, 1, 3, 1, 3, 3, 3
2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 2
1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3
4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 2, 2, 4
3, 3, 3, 1, 5, 3, 3, 5, 5, 3, 3, 5, 3, 3, 3, 1, 5, 3, 3, 5, 3, 3, 3, 1, 3
2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2
3, 1, 3, 3, 3, 5, 5, 3, 3, 5, 5, 3, 3, 1, 3, 3, 3, 5, 5, 3, 3, 1, 3, 3, 3
2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 2, 2, 4, 4, 2
1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
3, 5, 5, 3, 3, 1, 3, 3, 5, 3, 3, 5, 3, 5, 5, 3, 5, 3, 3, 5, 3, 5, 5, 3, 3
4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
5, 3, 3, 5, 3, 3, 3, 1, 5, 5, 5, 3, 5, 3, 5, 5, 5, 5, 5, 3, 5, 3, 5, 5, 5
4, 4, 4, 4, 4, 4, 2, 2, 6, 6, 4, 4, 4, 4, 6, 6, 6, 6, 4, 4, 4, 4, 6, 6, 4
3, 3, 3, 3, 3, 3, 1, 3, 5, 5, 3, 5, 3, 5, 5, 5, 5, 5, 3, 5, 3, 5, 5, 5, 3
2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2
MATHEMATICA
A101080[n_, k_]:= DigitCount[BitXor[n, k], 2, 1]; A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m= A006068[Floor[n/2]]}, 2m + Mod[Mod[n, 2] + Mod[m, 2], 2]]]; a[r_, 0]:= 0; a[0, c_]:=c; a[r_, c_]:= A003188[1 + A006068[a[r - 1, c - 1]]]; A[r_, c_]:= A101080[c, a[r, r + c]]; Table[A[c, r - c], {r, 0, 20}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 02 2017 *)
PROG
(Scheme)
(define (A268833bi row col) (A101080bi col (A268820bi row (+ row col))))
(PARI) b(n) = if(n<1, 0, b(n\2) + n%2);
for(r=0, 20, for(c=0, r, print1(A(c, r - c), ", "); ); print(); ) \\ Indranil Ghosh, Apr 02 2017
(PARI)
up_to = 32895; \\ = binomial(1+256, 2)-1.\\ A003188 and A006068 as above.
A268833list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A268833sq(a-col, col))); (v); };
v268833 = A268833list(1+up_to);
(Python)
def A101080(n, k): return bin(n^k)[2:].count("1")
if n<2: return n
else:
return 2*m + (n%2 + m%2)%2
for r in range(21):
print([a(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 02 2017
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
COMMENTS
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
FORMULA
Other identities. For all n >= 0:
MATHEMATICA
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 *)
PROG
(PARI)
A034386(n) = prod(i=1, primepi(n), prime(i));
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
(PARI)
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
CROSSREFS
Cf. A000975, A001222, A006068, A007913, A019565, A046523, A048675, A064707, A209281, A283477, A284004.
A permutation of the integers (a fractal sequence): a(n) = A006068(n-1) + 1.
+20
7
1, 2, 4, 3, 8, 7, 5, 6, 16, 15, 13, 14, 9, 10, 12, 11, 32, 31, 29, 30, 25, 26, 28, 27, 17, 18, 20, 19, 24, 23, 21, 22, 64, 63, 61, 62, 57, 58, 60, 59, 49, 50, 52, 51, 56, 55, 53, 54, 33, 34, 36, 35, 40, 39, 37, 38, 48, 47, 45, 46, 41, 42, 44, 43, 128, 127, 125, 126, 121, 122
EXAMPLE
Third nesting gives {1,2,4,3, 8,7,5,6} by means of joining the lists {1,2,4,3} = second nesting and {8,7,6,5} permuted by {1,2,4,3} giving {8,7,5,6}.
MATHEMATICA
Nest[ Join[ #, (Length[ #] + Range[ Length[ #], 1, -1 ])[[ # ]]] &, {1}, 7 ]
GrayCode[n_] := BitXor[n, Floor[n/2]]; t = Array[ GrayCode, 1000, 0]; Table[ Position[ t, n], {n, 0, 100}] // Flatten (* Robert G. Wilson v, Jun 22 2014 *)
PROG
(Python)
k, m = n-1, n-1>>1
while m > 0:
k ^= m
m >>= 1
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