A292385 -id:A292385 - cp's OEIS frontend

A292585 Restricted growth sequence transform of A278222(A292385(n)).

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

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

Term a(n) essentially records the run lengths of numbers of form 4k+1 encountered when starting from that node in binary tree A005940 which contains n, and by then traversing towards the root by iterating the map n -> A252463(n). The actual run lengths can be read from the exponents of primes in the prime factorization of A278222(A292385(m)), where m = min_{k=1..n} for which a(k) = a(n). In compound filter A292584 this is combined with similar information about the run lengths of the numbers of the form 4k+3 (A292583).

			When traversing from the root of binary tree A005940 from the node which contains 5, one obtains path 5 -> 3 -> 2 -> 1. Of these numbers, 5 and 1 are of the form 4k+1, while others are not, thus there are two separate runs of length 1: [1, 1]. On the other hand, when traversing from 9 as 9 -> 4 -> 2 -> 1, again only two terms are of the form 4k+1: 9 and 1 and they are not next to each other, so we have the same two runs of one each: [1, 1]. Similarly for n = 7, and n = 10 as neither in path 7 -> 5 -> 3 -> 2 -> 1 nor in path 10 -> 5 -> 3 -> 2 -> 1 are any more 4k+1 terms (compared to the path beginning from 5). Thus a(5), a(7), a(9) and a(10) are all allotted the same value by the restricted growth sequence transform, which in this case is 3.  Note that 3 occurs in this sequence for the first time at n=5, with A292385(5) = 5 and A278222(5) = 6 = 2^1 * 3^1, where those run lengths 1 and 1 are the prime exponents of 6.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A252463, A278222, A292375, A292385, A292583, A292584.

allocatemem(2^30);
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A278222(n) = A046523(A005940(1+n));
A252463(n) = if(!(n%2),n/2,A064989(n));
A292385(n) = if(n<=2,n-1,(if(1==(n%4),1,0)+(2*A292385(A252463(n)))));
write_to_bfile(1,rgs_transform(vector(16384,n,A278222(A292385(n)))),"b292585_upto16384.txt");

A163511 a(0)=1. a(n) = p(A000120(n)) * Product_{m=1..A000120(n)} p(m)^A163510(n,m), where p(m) is the m-th prime.

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 25, 12, 15, 10, 7, 32, 81, 54, 125, 36, 75, 50, 49, 24, 45, 30, 35, 20, 21, 14, 11, 64, 243, 162, 625, 108, 375, 250, 343, 72, 225, 150, 245, 100, 147, 98, 121, 48, 135, 90, 175, 60, 105, 70, 77, 40, 63, 42, 55, 28, 33, 22, 13, 128

Offset: 0

Leroy Quet, Jul 29 2009

This is a permutation of the positive integers.
From Antti Karttunen, Jun 20 2014: (Start)
Note the indexing: the domain starts from 0, while the range excludes zero, thus this is neither a bijection on the set of nonnegative integers nor on the set of positive natural numbers, but a bijection from the former set to the latter.
Apart from that discrepancy, this could be viewed as yet another "entanglement permutation" where the two complementary pairs to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with the complementary pair even numbers (taken straight) and odd numbers in the order they appear in A003961: (A005843/A003961). See also A246375 which has almost the same recurrence.
Note how the even bisection halved gives the same sequence back. (For a(0)=1, take ceiling of 1/2).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
This irregular table can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A003961 to the parent:
                                     1
                                     |
                  ...................2...................
                 4                                       3
       8......../ \........9                   6......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       27         18       25         12       15         10       7
32  81   54  125    36  75   50  49     24  45   30  35     20  21   14 11
etc.
Sequence A005940 is obtained by scanning the same tree level by level in mirror image fashion. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees, and A252463 gives the parent of the node containing n.
A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 1 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is smaller than the right child, and A252744(n) is an indicator function for those nodes.
(End)
Note that the idea behind maps like this (and the mirror image A005940) admits also using alternative orderings of primes, not just standard magnitude-wise ordering (A000040). For example, A332214 is a similar sequence but with primes rearranged as in A332211, and A332817 is obtained when primes are rearranged as in A108546. - Antti Karttunen, Mar 11 2020
From Lorenzo Sauras Altuzarra, Nov 28 2020: (Start)
This sequence is generated from A228351 by applying the following procedure: 1) eliminate the compositions that end in one unless the first one, 2) subtract one unit from every component, 3) replace every tuple [t_1, ..., t_r] by Product_{k=1..r} A000040(k)^(t_k) (see the examples).
Is it true that a(n) = A337909(n+1) if and only if a(n+1) is not a term of A161992?
Does this permutation have any other cycle apart from (1), (2) and (6, 9, 16, 7)? (End)
From Antti Karttunen, Jul 25 2023: (Start)
(In the above question, it is assumed that the starting offset would be 1 instead of 0).
Questions:
Does a(n) = 1+A054429(n) hold only when n is of the form 2^k times 1, 3 or 7, i.e., one of the terms of A029748?
It seems that A007283 gives all fixed points of map n -> a(n), like A335431 seems to give all fixed points of map n -> A332214(n). Is there a general rule for mappings like these that the fixed points (if they exist) must be of the form 2^k times a certain kind of prime, i.e., that any odd composite (times 2^k) can certainly be excluded? See also note in A029747.
(End)
If the conjecture given in A364297 holds, then it implies the above conjecture about A007283. See also A364963. - Antti Karttunen, Sep 06 2023
Conjecture: a(n^k) is never of the form x^k, for any integers n > 0, k > 1, x >= 1. This holds at least for squares, cubes, seventh and eleventh powers (see A365808, A365801, A366287 and A366391). - Antti Karttunen, Sep 24 2023, Oct 10 2023.
See A365805 for why the above holds for any n^k, with k > 1. - Antti Karttunen, Nov 23 2023

			For n=3, whose binary representation is "11", we have A000120(3)=2, with A163510(3,1) = A163510(3,2) = 0, thus a(3) = p(2) * p(1)^0 * p(2)^0 = 3*1*1 = 3.
For n=9, "1001" in binary, we have A000120(9)=2, with A163510(9,1) = 0 and A163510(9,2) = 2, thus a(9) = p(2) * p(1)^0 * p(2)^2 = 3*1*9 = 27.
For n=10, "1010" in binary, we have A000120(10)=2, with A163510(10,1) = 1 and A163510(10,2) = 1, thus a(10) = p(2) * p(1)^1 * p(2)^1 = 3*2*3 = 18.
For n=15, "1111" in binary, we have A000120(15)=4, with A163510(15,1) = A163510(15,2) = A163510(15,3) = A163510(15,4) = 0, thus a(15) = p(4) * p(1)^0 * p(2)^0 * p(3)^0 * p(4)^0 = 7*1*1*1*1 = 7.
[1], [2], [1,1], [3], [1,2], [2,1] ... -> [1], [2], [3], [1,2], ... -> [0], [1], [2], [0,1], ... -> 2^0, 2^1, 2^2, 2^0*3^1, ... = 1, 2, 4, 3, ... - _Lorenzo Sauras Altuzarra_, Nov 28 2020

Inverse: A243071.
Cf. A000040, A000120, A000225, A000788, A003961, A007814, A054429, A055396, A064216, A135523, A161992, A163510, A245605, A245612, A246375, A246378, A246681, A161511, A228351, A243499, A243503, A243504, A269854, A280873, A285727, A290251, A293437, A337909.
Cf. A007283 (known positions where a(n)=n), A029747, A029748, A364255 [= gcd(n,a(n))], A364258 [= a(n)-n], A364287 (where a(n) < n), A364292 (where a(n) <= n), A364494 (where n|a(n)), A364496 (where a(n)|n), A364963, A364297.
Cf. A365808 (positions of squares), A365801 (of cubes), A365802 (of fifth powers), A365805 [= A052409(a(n))], A366287, A366391.
Cf. A005940, A332214, A332817, A366275 (variants).

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~
Table[Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}]][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2016 *)

from sympy import prime
def A163511(n):
    if n:
        k, c, m = n, 0, 1
        while k:
            c += 1
            m *= prime(c)**(s:=(~k&k-1).bit_length())
            k >>= s+1
        return m*prime(c)
    return 1 # Chai Wah Wu, Jul 17 2023

For n >= 1, a(2n) is even, a(2n+1) is odd. a(2^k) = 2^(k+1), for all k >= 0.
From Antti Karttunen, Jun 20 2014: (Start)
a(0) = 1, a(1) = 2, a(2n) = 2*a(n), a(2n+1) = A003961(a(n)).
As a more general observation about the parity, we have:
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [This permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, A055396(a(n)) = A091090(n) = A007814(n+1) + 1 - A036987(n).
For n >= 1, a(A000225(n)) = A000040(n).
(End)
From Antti Karttunen, Oct 11 2014: (Start)
As a composition of related permutations:
a(n) = A005940(1+A054429(n)).
a(n) = A064216(A245612(n))
a(n) = A246681(A246378(n)).
Also, for all n >= 0, it holds that:
A161511(n) = A243503(a(n)).
A243499(n) = A243504(a(n)).
(End)
More linking identities from Antti Karttunen, Dec 30 2017: (Start)
A046523(a(n)) = A278531(n). [See also A286531.]
A278224(a(n)) = A285713(n). [Another filter-sequence.]
A048675(a(n)) = A135529(n) seems to hold for n >= 1.
A250245(a(n)) = A252755(n).
A252742(a(n)) = A252744(n).
A245611(a(n)) = A253891(n).
A249824(a(n)) = A275716(n).
A292263(a(n)) = A292264(n). [A292944(n) + A292264(n) = n.]
--
A292383(a(n)) = A292274(n).
A292385(a(n)) = A292271(n). [A292271(n) +  A292274(n) = n.]
--
A292941(a(n)) = A292942(n).
A292943(a(n)) = A292944(n).
A292945(a(n)) = A292946(n). [A292942(n) + A292944(n) + A292946(n) = n.]
--
A292253(a(n)) = A292254(n).
A292255(a(n)) = A292256(n). [A292944(n) + A292254(n) + A292256(n) = n.]
--
A279339(a(n)) = A279342(n).
a(A071574(n)) = A269847(n).
a(A279341(n)) = A279338(n).
a(A252756(n)) = A250246(n).
(1+A008836(a(n)))/2 = A059448(n).
(End)
From Antti Karttunen, Jul 26 2023: (Start)
For all n >= 0, a(A007283(n)) = A007283(n).
A001222(a(n)) = A290251(n).
(End)

More terms computed and examples added by Antti Karttunen, Jun 20 2014

A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 255, 28, 29, 62, 511, 24, 11, 126, 9, 60, 1023, 26, 2047, 16, 61, 254, 27, 20, 4095, 510, 125, 56, 8191, 58, 16383, 124, 25, 1022, 32767, 48, 23, 22, 253, 252, 65535, 18, 59, 120, 509, 2046, 131071

Offset: 1

Antti Karttunen, Jun 20 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

See also the comments at A163511, which is the inverse permutation to this one.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A163511.
Cf. A000040, A000225, A007814, A054429, A064989, A064216, A122111, A209229, A245611 (= (a(2n-1)-1)/2, for n > 1), A245612, A292383, A292385, A297171 (Möbius transform).
Cf. A007283 (known positions where a(n)=n), A364256 [= gcd(n,a(n))], A364288 [= n-a(n)], A364289 [where a(n)>=n], A364290 [where a(n)A364291 [where a(n)<=n], A364497 [where n|a(n)].
Cf. A156552 (variant with inverted binary code), A253566, A332215, A332811, A334859 (other variants).

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n))))); \\ Antti Karttunen, Jul 18 2020

A243071(n) = if(n<=2, n-1, my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p*p2*(2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429) - Antti Karttunen, Jul 28 2023

from functools import reduce
from sympy import factorint, prevprime
from operator import mul
def a064989(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, (1 if i==2 else prevprime(i)**f[i] for i in f))
def a(n): return n - 1 if n<3 else 2*a(n//2) if n%2==0 else 1 + 2*a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; With memoizing definec-macro from Antti Karttunen's IntSeq-library.
(definec (A243071 n) (cond ((<= n 2) (- n 1)) ((even? n) (* 2 (A243071 (/ n 2)))) (else (+ 1 (* 2 (A243071 (A064989 n)))))))

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A064989(2n+1)).
For n >= 1, a(A000040(n)) = A000225(n).
For n >= 1, a(2n+1) = 1 + 2*a(A064216(n+1)).
From Antti Karttunen, Jul 18 2020: (Start)
a(n) = A245611(A048673(n)).
a(n) = A253566(A122111(n)).
a(n) = A334859(A225546(n)).
For n >= 2, a(n) = A054429(A156552(n)).
a(n) = A292383(n) + A292385(n) = A292383(n) OR A292385(n).
For n > 1, A007814(a(n)) = A007814(n) - A209229(n). [This map  preserves the 2-adic valuation of n, except when n is a power of two, in which cases it is decremented by one.]
(End)

A292383 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 5, 0, 0, 4, 11, 4, 22, 10, 5, 0, 44, 0, 89, 8, 8, 22, 179, 8, 0, 44, 1, 20, 358, 10, 717, 0, 20, 88, 11, 0, 1434, 178, 45, 16, 2868, 16, 5737, 44, 8, 358, 11475, 16, 0, 0, 89, 88, 22950, 2, 17, 40, 176, 716, 45901, 20, 91802, 1434, 17, 0, 40, 40, 183605, 176, 356, 22, 367211, 0, 734422, 2868, 1, 356, 22, 90, 1468845, 32, 0, 5736, 2937691, 32

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 3, the starting value is of the form 4k+3, after which follows A252463(3) = 2, and A252463(2) = 1, the end point of iteration, and neither 2 nor 1 is of the form 4k+3, thus a(3) = 1*(2^0) + 0*(2^1) + 0*(2^2) = 1.
For n = 5, the starting value is not of the form 4k+3, after which follows A252463(5) = 3 (which is), continuing as before as 3 -> 2 -> 1, thus a(5) = 0*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 2.
For n = 10, the starting value is not of the form 4k+3, after which follows A252463(10) = 5 (also not 4k+3), and then A252463(5) = 3 (which is), continuing as before as 3 -> 2 -> 1, thus a(10) = 0*(2^0) + + 0*(2^1) + 1*(2^2) + 0*(2^3) + 0*(2^4) = 4.

Cf. A163511, A243071, A292274, A292373, A292377, A292380, A292381, A292382, A292384, A292385.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 3, 1, 0], 2], {n, 84}] (* Michael De Vlieger, Sep 21 2017 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A292383(n) = if(1==n,0,(if(3==(n%4),1,0)+(2*A292383(A252463(n)))));

(define (A292383 n) (A292373 (A292384 n)))

a(1) = 0; for n > 1, a(n) = 2*a(A252463(n)) + [n ≡ 3 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+3, and 0 otherwise.
a(n) = A292373(A292384(n)).
a(n) = A292274(A243071(n)).
Other identities. For n >= 1:
a(2n) = 2*a(n).
a(n) + A292385(n) = A243071(n).
a(A163511(n)) = A292274(n).
A000120(a(n)) = A292377(n).

A292381 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

1, 2, 4, 4, 9, 8, 18, 8, 9, 18, 36, 16, 73, 36, 16, 16, 147, 18, 294, 36, 37, 72, 588, 32, 19, 146, 16, 72, 1177, 32, 2354, 32, 73, 294, 32, 36, 4709, 588, 144, 72, 9419, 74, 18838, 144, 33, 1176, 37676, 64, 39, 38, 292, 292, 75353, 32, 74, 144, 589, 2354, 150706, 64, 301413, 4708, 72, 64, 147, 146, 602826, 588, 1177, 64, 1205652, 72, 2411305, 9418, 36, 1176

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 1, the starting value (which is also the ending point) is of the form 4k+1, thus a(1) = 1*(2^0) = 1.
For n = 2, the starting value is not of the form 4k+1, but its parent, A252463(2) = 1 is, thus a(2) = 0*(2^0) + 1*(2^1) = 2.
For n = 3, the starting value is not of the form 4k+1, after which follows 2 (also not 4k+1), and then 2 -> 1, and it is only the end-point of iteration which is of the form 4k+1, thus a(3) = 0*(2^0) + 0*(2^1) + 1*(2^2) = 4.
For n = 5, the starting value is of the form 4k+1, after which follows A252463(5) = 3 (which is not), and then continuing as before as 3 -> 2 -> 1, thus a(5) = 1*(2^0) + 0*(2^1) + 0*(2^2) + 1*(2^3) = 9.

Cf. A252463, A292371, A292375, A292380, A292382, A292383, A292384, A292385.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 1, 1, 0], 2], {n, 76}] (* Michael De Vlieger, Sep 21 2017 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A292381(n) = if(1==n,n,(if(1==(n%4),1,0)+(2*A292381(A252463(n)))));

from sympy.core.cache import cacheit
from sympy.ntheory.factor_ import digits
from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a292371(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join(['1' if i==1 else '0' for i in k]), 2)
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4
def a(n): return a292371(a292384(n))
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A292381 n) (A292371 (A292384 n)))

a(1) = 1; for n > 1, a(n) = 2*a(A252463(n)) + [n ≡ 1 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+1, and 0 otherwise.
a(n) = A292371(A292384(n)).
Other identities. For n >= 1:
a(2n) = 2*a(n).
A000120(a(n)) = A292375(n).
For n >= 2, a(n) = A004754(A292385(n)).

A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 2, 5, 0, 0, 1, 0, 0, 1, 12, 6, 7, 14, 0, 1, 0, 4, 1, 8, 10, 3, 0, 0, 21, 24, 0, 1, 28, 2, 3, 2, 0, 1, 0, 0, 5, 2, 2, 1, 22, 24, 17, 0, 12, 33, 32, 14, 35, 42, 28, 45, 24, 0, 1, 16, 2, 11, 48, 0, 59, 0, 8, 3, 0, 2, 5, 0, 16, 1, 4, 20, 3, 6, 6, 7, 8, 0, 1, 56, 0, 3, 0, 42, 5, 0, 48, 5, 0, 0, 1, 14, 2, 65, 64, 56, 49, 44, 4, 49, 64, 6, 57, 0

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 3, the starting value is a multiple of three, after which follows A253889(3) = 1, the end point of iteration, which is not a multiple of three, thus a(3) = 1*(2^0) = 1.
For n = 8, the starting value is not a multiple of three, after which follows A253889(8) = 3, which is, thus a(8) = 0*(2^0) + 1*(2^1) = 2.
For n = 9, the starting value is a multiple of three, after which follows A253889(9) = 8 (which is not), while A253889(8) = 3 (which is), thus a(9) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.

Cf. A048673, A064216, A253889, A291770, A292243, A292247.

Cf. also A292245, A292246, and A292381, A292383, A292385, and A292590, A292591 for similarly constructed sequences, and also A292250.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1];g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n;Table[FromDigits[#, 2] &@ Map[Boole[Divisible[#, 3]] &,  Reverse@ NestWhileList[Floor@ g[Floor[f[#]/2]] &, n, # > 1 &]], {n, 109}] (* Michael De Vlieger, Sep 16 2017 *)

(define (A292244 n) (A291770 (A292243 n)))

a(n) = A291770(A292243(n)).
Other identities. For all n >= 1:
a(A048673(n)) = A292247(n).
a(n) + A292245(n) = A064216(n).
a(n) AND A292245(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).

A292255 a(1) = 0, and for n > 1, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == -1], where J is the Jacobi-symbol.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 0, 2, 6, 0, 12, 6, 0, 0, 25, 0, 51, 4, 4, 12, 102, 0, 0, 24, 0, 12, 205, 0, 411, 0, 12, 50, 0, 0, 822, 102, 24, 8, 1645, 8, 3291, 24, 0, 204, 6582, 0, 0, 0, 48, 48, 13165, 0, 9, 24, 100, 410, 26330, 0, 52660, 822, 8, 0, 25, 24, 105321, 100, 204, 0, 210642, 0, 421284, 1644, 0, 204, 1, 48, 842569, 16, 0, 3290, 1685138, 16, 48, 6582

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+5 or of the form 12k+7 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

The AND - XOR formulas just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292253, A292256, A292263, A292943.

Cf. also A292941, A292945.

(define (A292255 n) (if (<= n 1) 0 (+ (if (and (odd? n) (= -1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292255 (A252463 n))))))

a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == -1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = -1, and 0 otherwise.
a(n) = A292263(n) AND (A292383(n) XOR A292945(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292385(n) XOR A292941(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292256(n).
For n >= 1, a(n) + A292253(n) + A292943(n) = A243071(n).

A292941 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 9, 4, 4, 8, 18, 8, 37, 18, 8, 8, 74, 8, 149, 16, 16, 36, 298, 16, 9, 74, 8, 36, 596, 16, 1193, 16, 36, 148, 16, 16, 2387, 298, 72, 32, 4774, 32, 9549, 72, 16, 596, 19098, 32, 19, 18, 148, 148, 38196, 16, 33, 72, 296, 1192, 76392, 32, 152785, 2386, 32, 32, 72, 72, 305571, 296, 596, 32, 611142, 32, 1222285, 4774, 16, 596, 32

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right (less significant end).

The AND - XOR formulas just restate the fact that J(-3|n) = J(-1|n)*J(3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292942, A292943, A292945.

Cf. also A292253, A292263, A292381, A292385.

(define (A292941 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 6)) 1 0) (* 2 (A292941 (A252463 n))))))

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+1, and 0 otherwise.
Also, for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = 1], where J is the Jacobi-symbol.
a(n) = A292263(n) AND (A292253(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292255(n) XOR A292385(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292942(n).
For n >= 1, a(n) + A292943(n) + A292945(n) = A243071(n).

cp's OEIS Frontend

A292271 a(n) = A292385(A163511(n)).

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A336125 a(n) = A292385(A122111(n)).

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A292585 Restricted growth sequence transform of A278222(A292385(n)).

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A163511 a(0)=1. a(n) = p(A000120(n)) * Product_{m=1..A000120(n)} p(m)^A163510(n,m), where p(m) is the m-th prime.

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A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A064989(2n+1)).

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A292383 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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A292381 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064989(2n+1)).

A292255 a(1) = 0, and for n > 1, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == -1], where J is the Jacobi-symbol.