A277006 -id:A277006 - cp's OEIS frontend

A277196 Permutation of natural numbers: a(n) = A107079(A277006(n)).

1, 2, 3, 4, 5, 6, 7, 11, 8, 10, 14, 23, 19, 9, 15, 21, 34, 28, 48, 44, 65, 12, 17, 26, 40, 41, 57, 69, 101, 89, 94, 144, 233, 129, 13, 22, 32, 53, 49, 75, 80, 120, 115, 111, 168, 279, 203, 137, 176, 261, 438, 283, 609, 470, 703, 16, 25, 35, 60, 63, 82, 104, 157, 128, 148, 217, 363, 236, 152, 227, 342, 569, 334, 798, 554, 833, 199, 270

Offset: 0

Antti Karttunen, Oct 07 2016

nonn

Note the indexing: domain starts from 0, but the range starts from 1.

Antti Karttunen, Table of n, a(n) for n = 0..353
Index entries for sequences that are permutations of the natural numbers

Inverse: A277195.

Cf. A000045, A003714, A005940, A071403, A107079, A277006.

(define (A277196 n) (A107079 (A277006 n)))

a(n) = A107079(A277006(n)) = A107079(A005940(1+A003714(n))).
Other identities:
For all n >= 1, a(A000045(n+1)) = A071403(n).

A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 5, 6, 16, 9, 10, 12, 7, 32, 17, 18, 20, 11, 24, 13, 14, 64, 33, 34, 36, 19, 40, 21, 22, 48, 25, 26, 28, 15, 128, 65, 66, 68, 35, 72, 37, 38, 80, 41, 42, 44, 23, 96, 49, 50, 52, 27, 56, 29, 30, 256, 129, 130, 132, 67, 136, 69, 70, 144, 73, 74, 76, 39, 160, 81

Offset: 0

Antti Karttunen

Permutation of the nonnegative integers (A001477); inverse permutation of A048680 i.e. A048679[ A048680[ n ] ] = n for all n.

Alois P. Heinz, Table of n, a(n) for n = 0..17710 (terms n = 10001..10945 from Antti Karttunen)
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A005203, A048678, A048680, A072650, A087808, A106151, A200714, A227351, A232559, A277006, A304100, A304101.

a(n) = rewrite_0to0_x1to1(fibbinary(j)) (where fibbinary(j) = A003714[ n ])
rewrite_0to0_x1to1 := proc(n) option remember; if(0 = n) then RETURN(n); else RETURN((2 * rewrite_0to0_x1to1(floor(n/(2^(1+(n mod 2)))))) + (n mod 2)); fi; end;
fastfib := n -> round((((sqrt(5)+1)/2)^n)/sqrt(5)); fibinv_appr := n -> floor(log[ (sqrt(5)+1)/2 ](sqrt(5)*n)); fibinv := n -> (fibinv_appr(n) + floor(n/fastfib(1+fibinv_appr(n)))); fibbinary := proc(n) option remember; if(n <= 2) then RETURN(n); else RETURN((2^(fibinv(n)-2))+fibbinary_seq(n-fastfib(fibinv(n)))); fi; end;
# second Maple program:
b:= proc(n) is(n=0) end:
a:= proc(n) option remember; local h; h:= iquo(a(n-1), 2)+1;
      while b(h) do h:= h*2 od; b(h):=true; h
    end: a(0):=0:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 22 2014

b[n_] := n==0; a[n_] := a[n] = Module[{h}, h = Quotient[a[n-1], 2] + 1; While[b[h], h = h*2]; b[h] = True; h]; a[0]=0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n))); \\ Antti Karttunen, May 13 2018, after Reinhard Zumkeller's May 09 2005 formula.

from itertools import count, islice
def A048679_gen(): # generator of terms
    return map(lambda n: int(bin(n)[2:].replace('01','1'),2),filter(lambda n:not (n<<1)&n,count(0)))
A048679_list = list(islice(A048679_gen(),20)) # Chai Wah Wu, Mar 18 2024

def A048679(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= n:
            s += 1
            n -= d
        else:
            s <<= 1
    return s # Chai Wah Wu, Apr 24 2025

a(n) = A106151(2*A003714(n)) for n > 0. - Reinhard Zumkeller, May 09 2005
a(n+1) = min{([a(n)/2]+1)*2^k} such that it is not yet in the sequence. - Gerard Orriols, Jun 07 2014
a(n) = A072650(A003714(n)) = A003188(A227351(n)). - Antti Karttunen, May 13 2018

A087808 a(0) = 0; a(2n) = 2a(n), a(2n+1) = a(n) + 1.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 4, 3, 8, 5, 6, 4, 8, 5, 6, 4, 16, 9, 10, 6, 12, 7, 8, 5, 16, 9, 10, 6, 12, 7, 8, 5, 32, 17, 18, 10, 20, 11, 12, 7, 24, 13, 14, 8, 16, 9, 10, 6, 32, 17, 18, 10, 20, 11, 12, 7, 24, 13, 14, 8, 16, 9, 10, 6, 64, 33, 34, 18, 36, 19, 20, 11, 40, 21, 22, 12

Offset: 0

Ralf Stephan, Oct 14 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

This is Guy Steele's sequence GS(5, 4) (see A135416); compare GS(4, 5): A135529.
A048678(k) is where k appears first in the sequence.
Cf. A000120, A003714, A004718, A005940, A048675, A048679, A080100, A090639.
A left inverse of A277020.
Cf. also A277006.

import Data.List (transpose)
a087808 n = a087808_list !! n
a087808_list = 0 : concat
   (transpose [map (+ 1) a087808_list, map (* 2) $ tail a087808_list])
-- Reinhard Zumkeller, Mar 18 2015

S := 2; f := proc(n) global S; option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(S*f(n/2)); else f((n-1)/2)+1; fi; end;

a[0]=0; a[n_] := a[n] = If[EvenQ[n], 2*a[n/2], a[(n-1)/2]+1]; Array[a,76,0] (* Jean-François Alcover, Aug 12 2017 *)

a(n)=if(n<1,0,if(n%2==0,2*a(n/2),a((n-1)/2)+1))

from functools import lru_cache
@lru_cache(maxsize=None)
def A087808(n): return 0 if n == 0 else A087808(n//2) + (1 if n % 2 else A087808(n//2)) # Chai Wah Wu, Mar 08 2022

(define (A087808 n) (cond ((zero? n) n) ((even? n) (* 2 (A087808 (/ n 2)))) (else (+ 1 (A087808 (/ (- n 1) 2)))))) ;; Antti Karttunen, Oct 07 2016

a(n) = A135533(n)+1-2^(A000523(n)+1-A000120(n)). - Don Knuth, Mar 01 2008
From Antti Karttunen, Oct 07 2016: (Start)
a(n) = A048675(A005940(n+1)).
For all n >= 0, a(A003714(n)) = A048679(n).
For all n >= 0, a(A277020(n)) = n.
(End)

A277010 a(n) = A156552(A005117(n)); permutation of Fibbinary numbers.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 32, 17, 10, 64, 128, 18, 33, 256, 65, 512, 21, 1024, 34, 129, 20, 2048, 257, 66, 4096, 37, 8192, 513, 16384, 130, 32768, 36, 258, 1025, 65536, 131072, 2049, 68, 69, 262144, 514, 41, 524288, 1048576, 4097, 40, 133, 2097152, 8193, 4194304, 132, 16385, 1026, 8388608, 72, 2050, 32769, 260, 16777216, 33554432, 261, 67108864, 42

Offset: 1

Antti Karttunen, Oct 07 2016

nonn

Permutation of A003714 (Fibbinary numbers).

Antti Karttunen, Table of n, a(n) for n = 1..1025

Cf. A000040, A003714, A005117, A071403, A156552.

Cf. also A277006, A277020, A277195.

from math import isqrt
from sympy import mobius, primepi, factorint
def A277010(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return sum(1<Chai Wah Wu, Aug 31 2024

(define (A277010 n) (A156552 (A005117 n)))

a(n) = A156552(A005117(n)).
Other identities. For all n >= 1:
a(A071403(n)) = A000079(n-1).

A280873 Numbers whose binary expansion does not begin 10 and does not contain 2 adjacent 0's; Ahnentafel numbers of X-chromosome inheritance of a male.

Original entry on oeis.org

0, 1, 3, 6, 7, 13, 14, 15, 26, 27, 29, 30, 31, 53, 54, 55, 58, 59, 61, 62, 63, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 213, 214, 215, 218, 219, 221, 222, 223, 234, 235, 237, 238, 239, 245, 246, 247, 250, 251, 253, 254, 255

Offset: 0

Floris Strijbos, Jan 09 2017

The number of ancestors at generation m from whom a living individual may have received an X chromosome allele is F_m, the m-th term of the Fibonacci Sequence.
From Antti Karttunen, Oct 11 2017: (Start)
The starting offset is zero (with a(0) = 0) for the same reason that we have A003714(0) = 0. Indeed, b(n) = A054429(A003714(n)) for n >= 0 yields the terms of this sequence, but in different order.
A163511(a(n)) for n >= 0 gives a permutation of squarefree numbers (A005117). See also A277006.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..10946
David Eppstein, Self-recursive generators (Python recipe)
L. A. D. Hutchison, N. M. Myres and S. R. Woodward, Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships, Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04, Colorado Springs), pp. 42-49, Sept. 2004.
Index entries for sequences related to binary expansion of n

Cf. A003714, A054429, A365538.
Intersection of A003754 and A004760.
Positions where A163511 obtains squarefree (A005117) values.
Cf. also A293437 (a subsequence).

gen[0]:= {0,1,3}:
gen[1]:= {6,7}:
for n from 2 to 10 do
  gen[n]:= map(t -> 2*t+1, gen[n-1]) union
      map(t -> 2*t, select(type, gen[n-1],odd))
od:
sort(convert(`union`(seq(gen[i],i=0..10)),list)); # Robert Israel, Oct 11 2017

male = {1, 3}; generations = 8;
Do[x = male[[i - 1]]; If[EvenQ[x],
                          male = Append[ male,   2*x + 1] ,
                          male = Flatten[Append[male, {2*x, 2*x + 1}]]]
       , {i, 3, Fibonacci[generations + 1]}]; male

isA003754(n) = { n=bitor(n, n>>1)+1; n>>=valuation(n, 2); (n==1); }; \\ After Charles R Greathouse IV's Feb 06 2017 code.
isA004760(n) = (n<2 || (binary(n)[2])); \\ This function also from Charles R Greathouse IV, Sep 23 2012
isA280873(n) = (isA003754(n) && isA004760(n));
n=0; k=0; while(k <= 10946, if(isA280873(n),write("b280873.txt", k, " ", n);k=k+1); n=n+1;); \\ Antti Karttunen, Oct 11 2017

def A280873():
    yield 1
    for x in A280873():
        if ((x & 1) and (x > 1)):
            yield 2*x
        yield 2*x+1
def take(n, g):
  '''Returns a list composed of the next n elements returned by generator g.'''
  z = []
  if 0 == n: return(z)
  for x in g:
    z.append(x)
    if n > 1: n = n-1
    else: return(z)
take(120, A280873())
# Antti Karttunen, Oct 11 2017, after the given Mathematica-code (by Floris Strijbos) and a similar generator-example for A003714 by David Eppstein (cf. "Self-recursive generators" link).

{a(n) : n >= 1} = {k >= 1 : A365538(A054429(k)) > 0}. - Peter Munn, Jan 22 2024

a(0) = 0 prepended and more descriptive alternative name added by Antti Karttunen, Oct 11 2017

A277332 a(n) = A253565(A003714(n)).

Original entry on oeis.org

1, 2, 3, 5, 9, 7, 25, 15, 11, 49, 35, 21, 75, 13, 121, 77, 55, 245, 33, 147, 105, 17, 169, 143, 91, 847, 65, 605, 385, 39, 363, 231, 165, 735, 19, 289, 221, 187, 1859, 119, 1183, 1001, 85, 845, 715, 455, 4235, 51, 507, 429, 273, 2541, 195, 1815, 1155, 23, 361, 323, 247, 3757, 209, 3179, 2431, 133, 2023, 1547, 1309, 13013, 95

Offset: 0

Antti Karttunen, Oct 12 2016

nonn

After the initial terms 1, 2 and 3, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 3, but with such a restriction that A253560 may not be applied twice in succession.
A permutation of A277334.
Note how A253565(A022340(n)) = A253565(2*A003714(n)) yields a permutation of A056911, odd squarefree numbers.

			55 = A253550(A253550(A253560(A253550(3)))), 55 is in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A003714, A022340, A253550, A253560, A253565.
Cf. A277334 (same sequence sorted into ascending order).
Cf. also A056911, A277006, A277331.

(define (A277332 n) (A253565 (A003714 n)))

a(n) = A253565(A003714(n)).

A277331 a(n) = A253563(A003714(n)).

Original entry on oeis.org

1, 2, 4, 8, 6, 16, 12, 18, 32, 24, 36, 54, 30, 64, 48, 72, 108, 60, 162, 90, 150, 128, 96, 144, 216, 120, 324, 180, 300, 486, 270, 450, 750, 210, 256, 192, 288, 432, 240, 648, 360, 600, 972, 540, 900, 1500, 420, 1458, 810, 1350, 2250, 630, 3750, 1050, 1470, 512, 384, 576, 864, 480, 1296, 720, 1200, 1944, 1080, 1800, 3000, 840

Offset: 0

Antti Karttunen, Oct 12 2016

nonn

After the initial terms 1, 2 and 4, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 4, but with such a restriction that A253550 may not be applied twice in succession.

A permutation of A055932.

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A003714, A055932 (same sequence sorted into ascending order), A253550, A253560, A253563, A122111.

Cf. also A277006, A277332.

(define (A277331 n) (A253563 (A003714 n)))

a(n) = A253563(A003714(n)).

a(n) = A122111(A277006(n)).

cp's OEIS Frontend

A277196 Permutation of natural numbers: a(n) = A107079(A277006(n)).

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A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

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A087808 a(0) = 0; a(2n) = 2a(n), a(2n+1) = a(n) + 1.

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A277010 a(n) = A156552(A005117(n)); permutation of Fibbinary numbers.

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A280873 Numbers whose binary expansion does not begin 10 and does not contain 2 adjacent 0's; Ahnentafel numbers of X-chromosome inheritance of a male.

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A277332 a(n) = A253565(A003714(n)).

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A277331 a(n) = A253563(A003714(n)).

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