A342707 T(n, k) is the result of replacing 2's by k's in the hereditary base-2 expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 1, 3, 3, 1, 0, 1, 2, 4, 4, 4, 1, 0, 2, 2, 5, 27, 5, 5, 1, 0, 0, 3, 6, 28, 256, 6, 6, 1, 0, 1, 1, 7, 30, 257, 3125, 7, 7, 1, 0, 0, 2, 8, 31, 260, 3126, 46656, 8, 8, 1, 0, 1, 2, 9, 81, 261, 3130, 46657, 823543, 9, 9, 1, 0

Offset

0,12

Links

Table of n, a(n) for n=0..77.

Wikipedia, Hereditary base-n notation

Formula

T(n, n) = A343255(n).

T(n, 0) = A345021(n).

T(n, 1) = A000120(n).

T(n, 2) = n.

T(n, 3) = A222112(n-1).

T(0, k) = 0.

T(1, k) = 1.

T(2, k) = k.

T(3, k) = k + 1.

T(4, k) = k^k = A000312(k).

T(5, k) = k^k + 1 = A014566(k).

T(6, k) = k^k + k = A066068(k).

T(7, k) = k^k + k + 1 = A066279(k).

T(16, k) = k^k^k = A002488(k).

T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

Example

Array T(n, k) begins:
  n\k|  0  1   2   3     4      5       6        7          8           9
  ---+-------------------------------------------------------------------
    0|  0  0   0   0     0      0       0        0          0           0
    1|  1  1   1   1     1      1       1        1          1           1
    2|  0  1   2   3     4      5       6        7          8           9
    3|  1  2   3   4     5      6       7        8          9          10
    4|  1  1   4  27   256   3125   46656   823543   16777216   387420489
    5|  2  2   5  28   257   3126   46657   823544   16777217   387420490
    6|  1  2   6  30   260   3130   46662   823550   16777224   387420498
    7|  2  3   7  31   261   3131   46663   823551   16777225   387420499
    8|  0  1   8  81  1024  15625  279936  5764801  134217728  3486784401
    9|  1  2   9  82  1025  15626  279937  5764802  134217729  3486784402
   10|  0  2  10  84  1028  15630  279942  5764808  134217736  3486784410

Prog

(PARI) T(n, k) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=k^T(e, k)); v }

Crossrefs

See A341907 for a similar sequence.

Cf. A000120, A000312, A002488, A014566, A066068, A066279, A222112, A343255, A345021.

Sequence in context: A358729 A001617 A333628 * A282947 A287200 A284387

Adjacent sequences: A342704 A342705 A342706 * A342708 A342709 A342710

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Jun 04 2021

Status

approved