A277905 - OEIS

A277905

Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 15, 20, 27, 36, 48, 64, 30, 40, 54, 72, 96, 128, 7, 25, 45, 60, 80, 81, 108, 144, 192, 256, 14, 50, 90, 120, 160, 162, 216, 288, 384, 512, 21, 28, 75, 100, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024, 42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048, 35, 63, 84, 112, 125, 225, 300, 400

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.

Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.

Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - Gus Wiseman, May 25 2024

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..166

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1; for n > 1, if A277896(a(n-1)) > 0, then a(n) = A277896(a(n-1)), otherwise a(n) = A019565(A277903(n)). [A naive recurrence for a one-dimensional version.]

Other identities. For all n >= 1:

A048675(a(n)) = A277903(n).

EXAMPLE

The irregular table begins as:

row terms

0 1;

1 2;

2 3, 4;

3 6, 8;

4 5, 9, 12, 16;

5 10, 18, 24, 32;

6 15, 20, 27, 36, 48, 64;

7 30, 40, 54, 72, 96, 128;

8 7, 25, 45, 60, 80, 81, 108, 144, 192, 256;

9 14, 50, 90, 120, 160, 162, 216, 288, 384, 512;

10 21, 28, 75, 100, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024;

11 42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048;

...

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Select[Range[0, 2^k], Total[2^(prix[#]-1)]==k&], {k, 0, 10}] (* Gus Wiseman, May 25 2024 *)

PROG

(Scheme)

(definec (A277905 n) (A277905bi (A277903 n) (A277904 n)))

(define (A277905bi row col) (let outloop ((k (A019565 row)) (col col)) (if (zero? col) k (let inloop ((j (+ 1 k))) (if (= (A048675 j) row) (outloop j (- col 1)) (inloop (+ 1 j))))))) ;; Very slow implementation.

;; Implementation based on a naive recurrence:

(definec (A277905 n) (if (= 1 n) n (let ((maybe_next (A277896 (A277905 (- n 1))))) (if (not (zero? maybe_next)) maybe_next (A019565 (A277903 n))))))

CROSSREFS

Cf. A019565 (the left edge, the only terms that are squarefree).

Cf. A000079 (the trailing edge).

Cf. A048675, A260443, A277886, A277896, A277903, A277904.

Row lengths are A018819 (number of partitions of binary rank n).

A000009 counts strict partitions, ranks A005117.

A029837 stc_sum or A070939 bin_len, opposite A070940 binexp_lastpos_1.

A048675 gives binary rank of prime indices, distinct A087207.

A048793 lists binary indices, product A096111, reverse A272020.

A061395 gives greatest prime index, least A055396.