cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A165199 a(n) is obtained by flipping every second bit in the binary representation of n starting at the second-most significant bit and on downwards.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 13, 12, 15, 14, 9, 8, 11, 10, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 53, 52, 55, 54, 49, 48, 51, 50, 61, 60, 63, 62, 57, 56, 59, 58, 37, 36, 39, 38, 33, 32, 35, 34, 45, 44, 47, 46, 41, 40, 43, 42, 106, 107, 104, 105, 110, 111, 108
Offset: 0

Views

Author

Leroy Quet, Sep 07 2009

Keywords

Comments

This is a self-inverse permutation of the positive integers.
Old name was: a(0) = 0, and for n>=1, let b(n,m) be the m-th digit, reading left to right, of binary n. (b(n, 1) is the most significant binary digit, which is 1.) Then a(n) is such that b(a(n),1)=1; and if b(n,m)=b(n,m-1) then b(a(n),m) does not = b(a(n),m-1); and if b(n,m) does not = b(n,m-1) then b(a(n), m) = b(a(n),m-1), for all m where 2 <= m <= number binary digits in n.
From Emeric Deutsch, Oct 06 2020: (Start)
a(n) is the index of the composition that is the conjugate of the composition with index n.
The index of a composition is defined to be the positive integer whose binary form has run-lengths (i.e., runs of 1's, runs of 0's, etc. from left to right) equal to the parts of the composition. Example: the composition 1,1,3,1 has index 46 since the binary form of 46 is 101110.
a(18) = 24. Indeed, since the binary form of 18 is 10010, the composition with index 18 is 1,2,1,1 (the run-lengths of 10010); the conjugate of 1,2,1,1 is 2,3 and so the binary form of a(18) is 11000; consequently, a(18) = 24. (End)

Examples

			a(12) = 9 because 12 = 1100_2 and 1100_2 XOR 0101_2 = 1001_2 = 9.

Links

Crossrefs

Cf. A000035, A000523, A075157, A075158, A241909, A243353, A245443, A245444.
{A001477, A129594, A165199, A056539} form a 4-group.

Programs

  • Maple
    a:= n-> Bits[Xor](n, iquo(2^(1+ilog2(n)), 3)):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 07 2020
  • PARI
    for(k=0,67,my(b(n)=vector(#digits(n,2),i,!(i%2)));print1(bitxor(k,fromdigits(b(k),2)),", ")) \\ Hugo Pfoertner, Oct 07 2020
  • PARI
    a(n) = if(n, bitxor(n,2<Kevin Ryde, Oct 07 2020
  • R
    maxrow <- 8 # by choice
a <- 1
for(m in 0: maxrow) for(k in 0:(2^m-1)){
a[2^(m+1) +       k] = a[2^(m+1) - 1 - k] + 2^(m+1)
a[2^(m+1) + 2^m + k] = a[2^(m+1) - 1 - k] + 2^m
}
(a <- c(0, a))
# Yosu Yurramendi, Apr 04 2017

Formula

From Antti Karttunen, Jul 22 2014: (Start)
a(0) = 0, and for n >= 1, a(n) = 2*a(floor(n/2)) + A000035(n+A000523(n)).
As a composition of related permutations:
a(n) = A056539(A129594(n)) = A129594(A056539(n)).
a(n) = A245443(A193231(n)) = A193231(A245444(n)).
a(n) = A075158(A243353(n)-1) = A075158((A241909(1+A075157(n))) - 1).
(End)
a(n) = A258746(A054429(n)) = A054429(A258746(n)), n > 0. - Yosu Yurramendi, Mar 29 2017

Extensions

Extended by Ray Chandler, Sep 10 2009
a(0) = 0 prepended by Antti Karttunen, Jul 22 2014
New name from Kevin Ryde, Oct 07 2020

A366260 Doudna sequence permuted by May code: a(n) = A005940(1+A303767(n)).

Original entry on oeis.org

1, 2, 4, 3, 9, 5, 6, 8, 16, 7, 10, 12, 15, 27, 25, 18, 54, 11, 14, 20, 21, 45, 35, 30, 24, 32, 49, 50, 36, 75, 81, 125, 625, 13, 22, 28, 33, 63, 55, 42, 40, 48, 77, 70, 60, 105, 135, 175, 90, 162, 121, 98, 100, 147, 225, 245, 150, 72, 64, 343, 250, 108, 375, 243, 729, 17, 26, 44, 39, 99, 65, 66, 56, 80, 91, 110
Offset: 0

Views

Author

Antti Karttunen, Oct 05 2023

Keywords

Links

Crossrefs

Cf. A005940, A303767, A366261, A366262.
Cf. also A243353, A366263.

Programs

A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 12, 14, 12, 84, 40, 44, 12, 142, 216, 183, 40, 265, 86, 152, 12, 142, 826, 265, 216, 1860, 607, 489, 40, 832, 607, 1117, 86, 619, 226, 560, 12, 142, 826, 265, 826, 5080, 2497, 619, 216, 2956, 4308, 4155, 607, 8575, 1105, 1533, 40, 832, 2497, 2116, 607, 5731, 4501, 3475, 86, 1402, 1105, 3475, 226, 1759, 698, 2144, 12, 142, 826, 265, 826, 5080, 2497, 619, 826
Offset: 0

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A003188, A278219, A278222, A286240, A286241, A286242.

Programs

  • PARI
    A003188(n) = bitxor(n, n>>1);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A278219(n) = A278222(A003188(n));
A286242(n) = (2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n))/2;
for(n=0, 16383, write("b286242.txt", n, " ", A286242(n)));
  • Python
    from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f=factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a003188(n): return n^(n>>1)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a278222(n): return a046523(a005940(n + 1))
def a(n): return T(a278222(n), a278219(n)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286242 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278219 n)) 2) (- (A278222 n)) (- (* 3 (A278219 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n)).
a(n) = (1/2)*(2 + ((A278222(n)+A278222(A003188(n)))^2) - A278222(n) - 3*A278222(A003188(n))).

A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 12, 14, 12, 59, 86, 27, 12, 109, 363, 269, 86, 142, 148, 27, 12, 109, 1093, 1117, 363, 1097, 1517, 489, 86, 601, 1408, 619, 148, 142, 148, 27, 12, 109, 1093, 1117, 1093, 5707, 8587, 2545, 363, 1969, 6153, 4529, 1517, 4489, 4537, 489, 86, 601, 3946, 3976, 1408, 2509, 5719, 2545, 148, 601, 1408, 619, 148, 142, 148, 27, 12, 109, 1093, 1117, 1093, 5707, 8587
Offset: 0

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A003188, A005940, A046523, A243353, A278219, A278222, A286240, A286242, A286255.

Programs

  • Mathematica
    f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; h[n_] := g@ f[BitXor[n, Floor[n/2]], 1, 1]; Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ # &, Table[{h[n], h[n + 1]}, {k, 12}, {n, k (k - 1)/2, k (k + 1)/2 - 1}]] // Flatten (* Michael De Vlieger, May 09 2017 *)
  • PARI
    A003188(n) = bitxor(n, n>>1);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A278219(n) = A278222(A003188(n));
A286241(n) = (2 + ((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n))/2;
for(n=0, 16383, write("b286241.txt", n, " ", A286241(n)));
  • Python
    from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a(n): return T(a278219(n), a278219(n + 1)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286241 n) (* (/ 1 2) (+ (expt (+ (A278219 n) (A278219 (+ 1 n))) 2) (- (A278219 n)) (- (* 3 (A278219 (+ 1 n)))) 2)))

Formula

a(n) = (1/2)*(2+((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n)).
Previous Showing 11-14 of 14 results.

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