A332898 -id:A332898 - cp's OEIS frontend

A332815 a(n) = A108548(A005940(1+n)).

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 13, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 11, 26, 39, 28, 65, 42, 63, 40, 91, 70, 105, 60, 175, 90, 135, 48, 169, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17, 22, 33, 52, 55, 78, 117, 56, 77, 130, 195, 84

Offset: 0

Antti Karttunen, Feb 28 2020

This is variant of Doudna-sequence, A005940 and thus can be represented as a binary tree. Each child to the left is obtained by applying A332818 to the parent, and each child to the right is obtained by doubling the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        5......../ \........6                   9......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    7       10         15       12         25       18         27       16
  13 14   21  20     35  30   45  24     49  50   75  36    125  54   81  32
etc.
Note the indexing: the sequence starts with a(0)=1, as is natural for sequences based on maps from base-2 expansion to prime factorization. This is
in contrast to A005940, which for historical reasons starts from offset 1.
For any n > 1, A332893(n) gives the value of the parent node. For any n >= 1, A332894(n) gives the distance to 1, and A332899(n) gives the number of odd numbers that occur (inclusively) on the path from 1 to n.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences that are permutations of the natural numbers

Cf. A332816 (inverse permutation).
Cf. A005940, A108548, A332818, A332893, A332894, A332897, A332898, A332899.
Cf. A108546 (the left edge of the tree from 2 downward).

up_to = 26927;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); };
A332815(n) = A108548(A005940(1+n));

a(n) = A108548(A005940(1+n)).

A332897 a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 2, 1, 1, 4, 2, 4, 2, 3, 3, 5, 1, 3, 3, 1, 2, 5, 1, 6, 1, 4, 4, 1, 2, 6, 4, 2, 2, 7, 3, 7, 3, 2, 5, 8, 1, 4, 3, 3, 3, 8, 1, 2, 2, 5, 5, 9, 1, 9, 6, 2, 1, 4, 4, 10, 4, 6, 1, 11, 2, 10, 6, 2, 4, 5, 2, 12, 2, 2, 7, 13, 3, 5, 7, 4, 3, 11, 2, 1, 5, 7, 8, 3, 1, 12, 4, 3, 3, 13, 3, 14, 3, 2

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Starting from x=n, iterate the map x -> A332893(x) which divides even numbers by 2, and for odd n changes every 4k+1 prime in their prime factorization to 4k+3 prime and vice versa (except 3 -> 2), like in A332819. a(n) counts the numbers of the form 4k+1 encountered until 1 has been reached, which is also included in the count when n > 1. This count includes also n itself when it is of the form 4k+1 (A016813) and larger than 1.

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

Cf. A016813, A332893, A332895, A332898, A332899.

Cf. also A292375.

A332897(n) = if(n<=2,n-1,A332897(A332893(n)) + (1==(n%4)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

a(n) = A000120(A332895(n)).

A332899 a(1) = 0, and for n > 2, a(n) = a(A332893(n)) + A000035(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 6, 2, 5, 4, 3, 1, 7, 2, 8, 3, 4, 6, 10, 2, 3, 5, 2, 4, 9, 3, 12, 1, 6, 7, 4, 2, 11, 8, 5, 3, 13, 4, 14, 6, 3, 10, 16, 2, 4, 3, 7, 5, 15, 2, 6, 4, 8, 9, 18, 3, 17, 12, 4, 1, 5, 6, 20, 7, 10, 4, 22, 2, 19, 11, 3, 8, 6, 5, 24, 3, 2, 13, 26, 4, 7, 14, 9, 6, 21, 3, 5, 10, 12, 16, 8, 2, 23, 4, 6, 3, 25, 7, 28, 5, 4

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

a(n) tells how many odd numbers are encountered when map x -> A332893(x) is used to traverse from n to 1, the root of the binary tree A332815. This count includes both the starting n itself if it is odd, but excludes 1 where the iteration ends.

a(n) also gives the index of the largest prime factor (A061395) in A332808(n), which is the inverse permutation of A108548 (see also A108546).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000035, A061395, A080791, A332808, A332811, A332815, A332816, A332893, A332894, A332897, A332898.

Cf. A000079 (after its initial term, gives the positions of 1's).

A332899(n) = if(1==n,0,A332899(A332893(n)) + (n%2));

a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + A000035(n).
a(n) = A000120(A332811(n)).
a(n) = A061395(A332808(n)).
a(n) = A332897(n) + A332898(n).
a(n) <= A332894(n).
For all n > 1, a(n) = 1 + A080791(A332816(n)).

A332894 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A332819(2n+1)); also binary width of terms of A332816.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 6, 4, 5, 5, 4, 4, 7, 4, 8, 5, 5, 7, 10, 5, 4, 6, 4, 6, 9, 5, 12, 5, 7, 8, 5, 5, 11, 9, 6, 6, 13, 6, 14, 8, 5, 11, 16, 6, 5, 5, 8, 7, 15, 5, 7, 7, 9, 10, 18, 6, 17, 13, 6, 6, 6, 8, 20, 9, 11, 6, 22, 6, 19, 12, 5, 10, 7, 7, 24, 7, 5, 14, 26, 7, 8, 15, 10, 9, 21, 6, 6, 12, 13, 17, 9, 7, 23, 6, 8, 6, 25, 9, 28, 8, 6

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

a(n) tells how many iterations of A332893 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A332815.

Each n > 0 occurs 2^(n-1) times in total.

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

A left inverse of A000079 and A108546.

Cf. A070939, A252464, A332808, A332815, A332816, A332819, A332893, A332897, A332898, A332899.

A332894(n) = if(1==n,0,if(!(n%2),1+A332894(n/2),1+A332894(A332819(n))));

A332894(n) = if(1==n,0,1+A332894(A332893(n)));

a(n) = A252464(A332808(n)).
a(1) = 0, and for n > 1, a(n) = 1 + a(A332893(n)).
For n >= 1, a(A108546(n)) = n; for all n >= 0, a(2^n) = n.
For n > 1: (Start)
a(n) = 1 + a(n/2) if n is even, and a(n) = 1 + a(A332819(n)), if n is odd.
a(n) = A070939(A332816(n)).
a(n) >= A332899(n).
(End)

A332901 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A332896(i)) = A278222(A332896(j)) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Restricted growth sequence transform of A278222(A332896(n)).
This is a variant of A292583: Instead of runs of numbers of the form 4k+3 encountered on trajectories of the standard Doudna-tree (A005940), this relates to the corresponding trajectories in A332815-tree. See comments in A292583.
For all i, j:
  a(i) = a(j) => A053866(i) = A053866(j),
  a(i) = a(j) => A332898(i) = A332898(j).

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

Cf. A053866, A278222, A332896, A332898, A332900.
Cf. A028982 (positions of ones).
Cf. also A292583.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v332901 = rgs_transform(vector(up_to,n,A278222(A332896(n))));
A332901(n) = v332901[n];

A332998 a(n) = A000120(A332996(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 05 2020

nonn

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

Cf. A000120, A332817, A332898, A332996, A332997.

Cf. A086224.

A332998(n) = hammingweight(A332996(n));

PARI
```
A332998(n) = A332898(A332817(n));
```

a(n) = A000120(A332996(n)) = A332898(A332817(n)).
a(n) = A000120(n) - A332997(n).
Conjecture: for n >= 2, a(A086224(n-1)) = n, this being also the first occurrence of each n larger than 1.

cp's OEIS Frontend

A332815 a(n) = A108548(A005940(1+n)).

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A332897 a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

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A332899 a(1) = 0, and for n > 2, a(n) = a(A332893(n)) + A000035(n).

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A332894 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A332819(2n+1)); also binary width of terms of A332816.

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A332901 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A332896(i)) = A278222(A332896(j)) for all i, j.

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A332998 a(n) = A000120(A332996(n)).

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