A334873 -id:A334873 - cp's OEIS frontend

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

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.

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.

Original entry on oeis.org

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];

A365387 a(n) = A331410(n) * A331410(A163511(n)).

Original entry on oeis.org

0, 0, 1, 0, 4, 1, 2, 0, 6, 4, 8, 1, 6, 2, 3, 0, 12, 6, 18, 4, 10, 8, 4, 1, 16, 6, 9, 2, 8, 3, 2, 0, 15, 12, 24, 6, 28, 18, 9, 4, 18, 10, 12, 8, 12, 4, 8, 1, 10, 16, 20, 6, 16, 9, 12, 2, 12, 8, 16, 3, 6, 2, 6, 0, 24, 15, 40, 12, 27, 24, 12, 6, 40, 28, 25, 18, 12, 9, 18, 4, 28, 18, 18, 10, 25, 12, 25, 8, 20, 12, 18, 4

Offset: 1

Antti Karttunen, Sep 03 2023

nonn

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

Cf. A163511, A331410, A365385.

Cf. also A334873.

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]))));
A365387(n) = (A331410(n)*A331410(A163511(n)));

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

cp's OEIS Frontend

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

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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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PARI

A365387 a(n) = A331410(n) * A331410(A163511(n)).

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PARI

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