A334204 -id:A334204 - cp's OEIS frontend

A334867 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A334204(i) = A334204(j) for all i, j >= 1.

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 6, 4, 6, 1, 7, 5, 8, 3, 8, 6, 9, 2, 5, 6, 8, 4, 8, 6, 10, 1, 11, 7, 9, 5, 9, 8, 12, 3, 13, 8, 14, 6, 11, 9, 15, 2, 15, 5, 13, 6, 9, 8, 9, 4, 15, 8, 16, 6, 8, 10, 17, 1, 12, 11, 14, 7, 14, 9, 18, 5, 11, 9, 19, 8, 20, 12, 21, 3, 14, 13, 12, 8, 22, 14, 21, 6, 12, 11, 14, 9, 14, 15, 15, 2, 23, 15, 14, 5, 11, 13, 12, 6, 14

Offset: 1

Antti Karttunen, Jun 09 2020

Restricted growth sequence transform of the ordered pair [A329697(n), A334204(n)].
For all i, j:
  A365388(i) = A365388(j) => a(i) = a(j) => A334873(i) = A334873(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079 (positions of ones), A163511, A329697, A334204, A334873.

Cf. also A318310, A365386, A365388.

up_to = 65537;
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334204(n) = A329697(A163511(n));
A334867aux(n) = [A329697(n),A334204(n)];
v334867 = rgs_transform(vector(up_to,n,A334867aux(n)));
A334867(n) = v334867[n];

A334873 a(n) = A329697(n) * A334204(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 6, 2, 4, 1, 4, 2, 4, 0, 4, 6, 9, 2, 9, 4, 12, 1, 6, 4, 9, 2, 9, 4, 6, 0, 15, 4, 12, 6, 12, 9, 18, 2, 8, 9, 20, 4, 15, 12, 16, 1, 16, 6, 8, 4, 12, 9, 12, 2, 16, 9, 12, 4, 9, 6, 8, 0, 18, 15, 20, 4, 20, 12, 32, 6, 15, 12, 21, 9, 28, 18, 24, 2, 20, 8, 18, 9, 12, 20, 24, 4, 18, 15, 20, 12, 20

Offset: 1

Antti Karttunen, Jun 09 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A163511, A329697, A334204, A334867.

Cf. A000079 (positions of zeros); It is also conjectured that A007283 gives the positions of ones, and that A070875 gives the indices of twos.

PARI
```
A334873(n) = (A329697(n) * A334204(n));
```

a(n) = A329697(n) * A334204(n) = A329697(n) * A329697(A163511(n)).

A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 0, 3, 1, 3, 2, 3, 3, 3, 1, 2, 3, 4, 2, 3, 3, 4, 1, 4, 2, 2, 2, 3, 3, 3, 2, 4, 3, 4, 2, 3, 3, 4, 0, 3, 3, 4, 1, 4, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 1, 4, 2, 3, 3, 2, 4, 4, 2, 3, 3, 4, 3, 4, 4, 4, 1, 2, 4, 4, 2

Offset: 1

Ali Sada and Robert G. Wilson v, Feb 28 2020

From Antti Karttunen, Apr 07 2020: (Start)
Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).
Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.
The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.
For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.
(End)
A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is {12, 8}, taking 2 iterations to reach 8 = 2^3. So a(15) is 2.
From _Antti Karttunen_, Apr 07 2020: (Start)
Considering all possible paths from 15 to 1 nondeterministic map k -> k-(k/p), where p can be any prime factor of k, we obtain the following graph:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.
It can be seen that there's also alternative route to 8 via 10 (with 10 = 15-(15/3), where 3 is not the largest prime factor of 15), but it's not any shorter than the route via 12.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree labeling the index n, with a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value in a(n) for n = 1..16383.

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for positions of 0 .. 6 in this sequence, and also array A334100.
Cf. A334099 (a right inverse, positions of the first occurrence of each n).
Cf. A334091 (first differences), A335429 (partial sums).
Cf. also A331410 (analogous sequence when using the map k -> k + k/p), A334861, A335877 (their sums and differences), see also A335878 and A335884, A335885.

a[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] -1; Array[a, 100]

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 07 2020

up_to = 2^24;
A329697list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = if(!bitand(n,n-1),0,1+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v329697 = A329697list(up_to);
A329697(n) = v329697[n]; \\ Antti Karttunen, Apr 07 2020

A329697(n) = if(n<=2,0, if(isprime(n), A329697(n-1)+1, my(f=factor(n)); (apply(A329697, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, Apr 19 2020

From Antti Karttunen, Apr 07-19 2020: (Start)
a(1) = a(2) = 0; and for n > 2, a(p) = 1 + a(p-1) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [This is otherwise equal to the definition of A064097, except here we have a different initial condition, with a(2) = 0].
a(2n) = a(A000265(n)) = a(n).
a(p) = 1+a(p-1), for all odd primes p.
If A209229(n) == 1 [when n is a power of 2], a(n) = 0,
  otherwise a(n) = 1 + a(n-A052126(n)) = 1 + a(A171462(n)).
Equivalently, for non-powers of 2, a(n) = 1 + a(n-(n/A078701(n))),
or equivalently, for non-powers of 2, a(n) = 1 + Min a(n - n/p), for p prime and dividing n.
a(n) = A064097(n) - A064415(n), or equally, a(n) = A064097(n) - A054725(n), for n > 1.
a(A019434(n)) = 1, a(A334092(n)) = 2, a(A334093(n)) = 3, etc. for all applicable n.
For all n >= 0, a(A334099(n)) = a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = a(257^n) = a(65537^n) = n.
a(A122111(n)) = A334107(n), a(A225546(n)) = A334109(n).
(End)
From Antti Karttunen, Mar 16 2021: (Start)
a(n) = a(A336466(n)) + A087436(n) = A336396(n) + A087436(n).
a(A053575(n)) = A336469(n) = a(n) - A005087(n).
a(A147545(n)) = A000120(A147545(n)) - 1.
(End)

A334860 a(0) = 1, a(1) = 2, after which, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 16, 5, 36, 18, 81, 12, 64, 32, 256, 10, 25, 72, 1296, 27, 324, 162, 6561, 24, 144, 128, 4096, 48, 1024, 512, 65536, 15, 100, 50, 625, 108, 5184, 2592, 1679616, 54, 729, 648, 104976, 243, 26244, 13122, 43046721, 20, 576, 288, 20736, 192, 16384, 8192, 16777216, 96, 2304, 2048, 1048576, 768, 262144, 131072, 4294967296, 30

Offset: 0

Antti Karttunen, Jun 08 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A334747 to the parent, and each child to the right is obtained by squaring the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        6......../ \........9                   8......../ \........16
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    5       36         18       81         12       64         32      256
  10 25   72  1296   27  324 162  6561   24  144  128 4096   48 1024 512 65536
etc.
This is the mirror image of the tree in A334866.
Fermi-Dirac primes, A050376, occur at rightward growing branches that originate from primes situated at the left edge.
The tree illustrated in A163511 is expanded as x -> 2*x for the left child and x -> A003961(x) for the right child, while this tree is expanded as x -> A225546(2*A225546(x)) for the left child, and x -> A225546(A003961(A225546(x))) for the right child.

Index entries for sequences that are permutations of the natural numbers

Cf. A000290, A225546, A334204, A334747, A334859 (inverse), A334866 (mirror image).
Cf. A001146 (right edge of the tree), A019565 (left edge), A334110 (the right children of the left edge).
Composition of permutations A163511 and A225546.

A334747(n) = { my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m; }; \\ From A334747
A334860(n) = if(n<=1,1+n,if(!(n%2),A334747(A334860(n/2)),A334860((n-1)/2)^2));

a(0) = 1, a(1) = 2; and for n > 0, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.
a(n) = A225546(A163511(n)).
For n >= 0, a(2^n) = A019565(1+n), a(2^((2^n)-1)) = A000040(1+n).
A334109(a(n)) = A334204(n).
It seems that for n >= 1, A048675(a(n)) = A135529(n) = A048675(A163511(n)).

A334109 a(n) = A329697(A225546(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 4, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 8, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 4, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 8, 2, 5, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Conjecture: Each k >= 0 occurs for the first time at A334110(k) = A019565(k)^2. Note that each k must occur first time on square n, because of the identity a(n) = a(A008833(n)). However, is there any reason to exclude squares with prime exponents > 2 from the candidates? See also comments in A334204.

Antti Karttunen, Table of n, a(n) for n = 1..10200
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001248, A005117 (positions of zeros), A334110.
Cf. A003961, A008833, A019565, A059895, A225546, A329697, A331590, A331736, A334204, A334860.
Cf. also A334107, A334108.

Map[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] ] (* Michael De Vlieger, May 26 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334109(n) = { my(f=factor(n),pis=apply(primepi,f[,1]),es=f[,2]); sum(k=1,#f~,(2^(pis[k]-1))*A329697(A019565(es[k]))); };

Additive with a(prime(i)^j) = A000079(i-1) * A329697(A019565(j)), a(m*n) = a(m)+a(n) if gcd(m,n) = 1.
Alternatively, additive with a(prime(i)^(2^k)) = 2^(i-1) * A329697(prime(k+1)), a(m*n) = a(m)+a(n) if A059895(m,n) = 1. - Peter Munn, May 04 2020
a(n) = A329697(A225546(n)) = A329697(A331736(n)).
a(n) = a(A008833(n)).
For all n >= 0, a(A334110(n)) = n, a(A334860(n)) = A334204(n).
a(A331590(m,k)) = a(m) + a(k); a(A003961(n)) = 2*a(n). - Peter Munn, Apr 30 2020

A365385 a(n) = A331410(A163511(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 2, 0, 3, 2, 4, 1, 3, 2, 1, 0, 4, 3, 6, 2, 5, 4, 2, 1, 4, 3, 3, 2, 2, 1, 2, 0, 5, 4, 8, 3, 7, 6, 3, 2, 6, 5, 4, 4, 3, 2, 4, 1, 5, 4, 5, 3, 4, 3, 3, 2, 3, 2, 4, 1, 3, 2, 2, 0, 6, 5, 10, 4, 9, 8, 4, 3, 8, 7, 5, 6, 4, 3, 6, 2, 7, 6, 6, 5, 5, 4, 5, 4, 4, 3, 6, 2, 5, 4, 4, 1, 6, 5, 7, 4, 6, 5, 4, 3, 5

Offset: 0

Antti Karttunen, Sep 03 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000265, A163511, A331410.
Cf. A131577 (positions of 0's).
Cf. also A334204, A364260.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
A365385(n) = A331410(A163511(n));

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A364260 a(n) = A331410(A332214(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 2, 0, 4, 3, 3, 2, 3, 2, 4, 1, 3, 2, 3, 1, 3, 2, 1, 0, 5, 4, 4, 3, 4, 3, 6, 2, 4, 3, 5, 2, 5, 4, 2, 1, 4, 3, 4, 2, 4, 3, 3, 1, 4, 3, 2, 2, 2, 1, 2, 0, 6, 5, 5, 4, 5, 4, 8, 3, 5, 4, 7, 3, 7, 6, 3, 2, 5, 4, 6, 3, 6, 5, 4, 2, 6, 5, 3, 4, 3, 2, 4, 1, 5, 4, 5, 3, 5, 4, 5, 2, 5

Offset: 0

Antti Karttunen, Jul 16 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A163511, A331410, A332214, A364259.
Cf. A131577 (positions of 0's), A335431 (of 1's).
Differs from related A334204 for the first time at n=31, where a(31) = 1, while A334204(31) = 2.

a(n) = A364259(A163511(n)).

For all n >= 0, a(2^n) = 0, a(2^n + 1) = n.

A365388 Lexicographically earliest infinite sequence such that a(i) = a(j) => A334867(i) = A334867(i) and A365386(j) = A365386(j) for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42, 11, 43, 22, 44, 6, 45, 23, 46, 12, 47, 24, 48, 2, 49, 25, 41

Offset: 1

Antti Karttunen, Sep 07 2023

nonn

Restricted growth sequence transform of the ordered pair [A334867(n), A365386(n)], or equally, of the quadruplet [A329697(n), A334204(n), A331410(n), A365385(n)].
For all i, j:
  A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A334867(i) = A334867(j),
  a(i) = a(j) => A335880(i) = A335880(j),
  a(i) = a(j) => A365386(i) = A365386(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A163511, A329697, A331410, A334204, A334867, A335880, A365385, A365386.

Differs from A003602 and A351090 for the first time at n=99, where a(99) = 41, while A003602(99) = A351090(99) = 50.

up_to = 65537;
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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334204(n) = A329697(A163511(n));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
A365385(n) = A331410(A163511(n));
A365388aux(n) = [A329697(n),A334204(n),A331410(n),A365385(n)];
v365388 = rgs_transform(vector(up_to,n,A365388aux(n)));
A365388(n) = v365388[n];

A335420 a(n) = A000120(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 4, 2, 3, 1, 3, 4, 6, 2, 4, 3, 3, 2, 4, 4, 3, 2, 3, 3, 3, 1, 6, 3, 5, 4, 7, 6, 6, 2, 4, 4, 6, 3, 4, 3, 5, 2, 4, 4, 6, 4, 4, 3, 4, 2, 6, 3, 5, 3, 2, 3, 3, 1, 6, 6, 6, 3, 7, 5, 5, 4, 5, 7, 7, 6, 3, 6, 6, 2, 5, 4, 5, 4, 8, 6, 7, 3, 6, 4, 6, 3, 6, 5, 4, 2, 5, 4, 7, 4, 4, 6, 5, 4, 6

Offset: 0

Antti Karttunen, Jun 09 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000079 (positions of ones), A000120, A001222, A163511, A335421, A335422.

Cf. also A323901, A334204.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A335420(n) = hammingweight(A163511(n));

from sympy import nextprime
def A335420(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return (c*p).bit_count() # Chai Wah Wu, Jul 25 2023

a(n) = A000120(A163511(n)).
a(n) = A001222(A335422(n)).
a(n) = a(2n) = a(A000265(n)).
For all n >= 0, a(2^n) = 1.

A365394 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365425(i) = A365425(j) and A365426(i) = A365426(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 5, 2, 6, 1, 7, 4, 4, 3, 8, 3, 9, 2, 8, 5, 10, 2, 10, 6, 11, 1, 12, 7, 7, 4, 13, 4, 14, 3, 15, 8, 16, 3, 16, 9, 17, 2, 13, 8, 18, 5, 19, 10, 20, 2, 18, 10, 21, 6, 21, 11, 6, 1, 22, 12, 12, 7, 23, 7, 24, 4, 25, 13, 26, 4, 26, 14, 27, 3, 25, 15, 28, 8, 29, 16, 30, 3, 28, 16, 31, 9, 31, 17

Offset: 0

Antti Karttunen, Sep 04 2023

Restricted growth sequence transform of the ordered pair [A365425(n), A365426(n)].
Restricted growth sequence transform of the function f(n) = A336470(A163511(n)).
For all i, j: a(i) = a(j) => A334204(i) = A334204(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000265, A003602, A046523, A163511, A336470, A365425, A365426.

Cf. also A350067, A365395, A366792 (compare the scatter plots).

up_to = 65537;
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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A365425(n) = A046523(A000265(A163511(n)));
A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };
A365426(n) = A336466(A163511(n));
A365394aux(n) = [A365425(n), A365426(n)];
v365394 = rgs_transform(vector(1+up_to,n,A365394aux(n-1)));
A365394(n) = v365394[1+n];

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

cp's OEIS Frontend

A334867 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A334204(i) = A334204(j) for all i, j >= 1.

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A334873 a(n) = A329697(n) * A334204(n).

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A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

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A334860 a(0) = 1, a(1) = 2, after which, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

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A334109 a(n) = A329697(A225546(n)).

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A365385 a(n) = A331410(A163511(n)).

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A364260 a(n) = A331410(A332214(n)).

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A365388 Lexicographically earliest infinite sequence such that a(i) = a(j) => A334867(i) = A334867(i) and A365386(j) = A365386(j) for all i, j >= 1.

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A335420 a(n) = A000120(A163511(n)).