A323505 Mirror image of (denominators of) Bernoulli tree, A106831.

1, 2, 4, 6, 8, 12, 12, 24, 16, 24, 24, 48, 24, 36, 48, 120, 32, 48, 48, 96, 48, 72, 96, 240, 48, 72, 72, 144, 96, 144, 240, 720, 64, 96, 96, 192, 96, 144, 192, 480, 96, 144, 144, 288, 192, 288, 480, 1440, 96, 144, 144, 288, 144, 216, 288, 720, 192, 288, 288, 576, 480, 720, 1440, 5040, 128, 192, 192, 384, 192, 288, 384, 960, 192, 288, 288

Offset

0,2

Comments

In contrast to A106831 which follows Woon's original indexing (and orientation), this variant starts with value a(0) = 1, with the rest of terms having an index incremented by one, thus allowing for a simple recurrence.

Sequence contains only terms of A001013 and each a(n) is a multiple of A246660(n).

Links

Antti Karttunen, Table of n, a(n) for n = 0..16383

S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.

Index entries for sequences related to Bernoulli numbers.

Formula

a(0) = 1; and for n > 0, if n is even, a(n) = 2*a(n/2), and if n is odd, a(n) = (A001511(n+1)+1-A036987(n)) * a((n-1)/2).

For n > 0, a(n) = b(A054429(n)), where b(n) = A106831(n-1).

a(n) = A246660(n) * A323506(n).

a(n) = A323508(A005940(1+n)).

Example

This sequence can be represented as a binary tree:
                                       1
                                       |
                    ...................2....................
                   4                                        6
         8......../ \........12                 12........./ \.......24
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       24         24       48         24       36         48      120
  32  48   48  96     48  72   96  240    48  72   72  144    96  144 240  720
etc.

Prog

(PARI)
A001511(n) = (1+valuation(n, 2));
A036987(n) = !bitand(n, 1+n);
A323505(n) = if(!n, 1, if(!(n%2), 2*A323505(n/2), (A001511(n+1)+1-A036987(n))*A323505((n-1)/2)));
(PARI)
A054429(n) = if(!n, n, ((3<<#binary(n\2))-n-1)); \\ From A054429
A106831r1(n) = if(!n, 1, if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Recurrence for A106831, when prepended with 1, thus shifted one term right
A323505(n) = A106831r1(A054429(n));

Crossrefs

Cf. A000079 (left edge), A000142 (right edge), A001013, A001511, A036987, A054429, A246660, A323506, A323508.

Cf. A106831 and also A005940, A283477, A322827 for other similar trees.

Sequence in context: A360408 A274262 A092990 * A350355 A172311 A103829

Adjacent sequences: A323502 A323503 A323504 * A323506 A323507 A323508

Keyword

nonn,tabf

Author

Antti Karttunen, Jan 16 2019

Status

approved