A334867 -id:A334867 - cp's OEIS frontend

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.

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];

A334204 a(n) = A329697(A163511(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, 2, 0, 5, 4, 4, 3, 4, 3, 6, 2, 4, 3, 5, 2, 5, 4, 4, 1, 4, 3, 4, 2, 4, 3, 4, 1, 4, 3, 3, 2, 3, 2, 2, 0, 6, 5, 5, 4, 5, 4, 8, 3, 5, 4, 7, 3, 7, 6, 6, 2, 5, 4, 6, 3, 6, 5, 6, 2, 6, 5, 5, 4, 5, 4, 4, 1, 5, 4, 5, 3, 5, 4, 6, 2, 5

Offset: 0

Antti Karttunen, Jun 09 2020

nonn

As the underlying sequence A163511 can be represented as a binary tree, so can be this:
                                     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 2
etc.
The nodes at the left edge are all zeros, and their right-hand children give positive integers, A000027.
Each left-hand leaning branch stays constant, because A329697(2n) = A329697(n).
The right-hand leaning branches are not necessarily monotonic. For example, a((2^6)-1) = 2 > 1 = a((2^7)-1), because A000040(7) = 17 is a Fermat prime (but A000040(6) = 13 is not), and therefore the latter is only one step away from a power of 2.

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

Cf. A000079, A163511, A329697, A334109, A334860, A334867, A334873.

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

a(n) = A329697(A163511(n)).
a(n) = A334109(A334860(n)).
a(n) = a(2n) = a(A000265(n)).
For all n >= 0, a(2^n) = 0, a(2^n + 1) = n.

A335880 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A331410(i) = A331410(j) for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 5, 2, 5, 4, 6, 1, 7, 5, 8, 3, 9, 5, 9, 2, 10, 5, 8, 4, 11, 6, 12, 1, 8, 7, 8, 5, 11, 8, 8, 3, 6, 9, 13, 5, 11, 9, 14, 2, 14, 10, 10, 5, 11, 8, 11, 4, 15, 11, 15, 6, 9, 12, 13, 1, 11, 8, 15, 7, 13, 8, 13, 5, 16, 11, 16, 8, 13, 8, 13, 3, 15, 6, 8, 9, 17, 13, 18, 5, 16, 11, 13, 9, 14, 14, 18, 2, 6, 14, 15, 10, 16, 10, 8, 5, 15

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

Restricted growth sequence transform of the ordered pair [A329697(n), A331410(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A334861(i) = A334861(j),
  a(i) = a(j) => A335875(i) = A335875(j),
  a(i) = a(j) => A335877(i) = A335877(j),
  a(i) = a(j) => A335881(i) = A335881(j).

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

Cf. A329697, A331410, A334861, A335875, A335877, A335881.

Cf. also A324400, A334867.

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; };
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
Aux335880(n) = [A329697(n),A331410(n)];
v335880 = rgs_transform(vector(up_to, n, Aux335880(n)));
A335880(n) = v335880[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)).

A365386 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A365385(i) = A365385(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 03 2023

Restricted growth sequence transform of the ordered pair [A331410(n), A365385(n)].

For all i, j: A365388(i) = A365388(j) => a(i) = a(j) => A365387(i) = A365387(j).

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

Cf. A003602, A163511, A331410, A365385, A365387, A365388.

Cf. also A334867.

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));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
A365385(n) = A331410(A163511(n));
A365386aux(n) = [A331410(n),A365385(n)];
v365386 = rgs_transform(vector(up_to,n,A365386aux(n)));
A365386(n) = v365386[n];

A366791 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366388(j) = A366388(j) and A366788(i) = A366788(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 23 2023

Restricted growth sequence transform of the ordered pair [A366388(n), A366788(n)].

For all i, j >= 1: A003602(i) = A003602(j) => a(i) = a(j).

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

Cf. A003602, A366388, A366788.

Cf. also A334867, A365386, A365388 (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; };
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));
A366388(n) = if(n<=2, 0, if(isprime(n), 1+A366388(primepi(n)), my(f=factor(n)); (apply(A366388, f[, 1])~ * f[, 2])));
Aux366791(n) = [A366388(n), A366388(A163511(n))];
v366791 = rgs_transform(vector(up_to, n, Aux366791(n)));
A366791(n) = v366791[n];

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A335880 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A331410(i) = A331410(j) for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A365386 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A365385(i) = A365385(j) for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A366791 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366388(j) = A366388(j) and A366788(i) = A366788(j), for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI