Riffs & Rotes

[Jon Awbrey]

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R&R.  Note 9

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SeqFans,

Here are some more terms of A108532 [pending],
the sequence of "primal code characteristics".

A108352 [pending]

1 0 2 2 2 0 2 2 0 0
2 0 2 0 2 2 2 0 2 3
2 0 2 3 2 0 2 3 2 0
2 2 2 0 2 0 2 0 2 0
2 0 2 3 0 0 2 3 2 0

a(n) = the primal code characteristic of n,
which is the least positive integer, if any,
such that (n o)^k = 1, otherwise equal to 0.

"o" = primal composition operation,
as illustrated in sequence A106177.

(n o)^k = n o ... o n (with k appearances of n).

Examples:
 a(1) = 1 because  (1 o)^1 = ({ } o)^1 = 1.
 a(2) = 0 because  (2 o)^k = (1:1 o)^k = 2, for all positive k.
 a(3) = 2 because  (3 o)^2 = (2:1 o)^2 = 1.
 a(4) = 2 because  (4 o)^2 = (1:2 o)^2 = 1.
 a(5) = 2 because  (5 o)^2 = (3:1 o)^2 = 1.
 a(6) = 0 because  (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
 a(7) = 2 because  (7 o)^2 = (4:1 o)^1 = 1.
 a(8) = 2 because  (8 o)^2 = (1:3 o)^1 = 1.
 a(9) = 0 because  (9 o)^k = (2:2 o)^k = 9, for all positive k.
a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.

a(11) = 2 because (11 o)^k = (5:1 o)^2 = 1.
a(12) = 0 because (12 o)^k in {12, 18}, for all positive k.
a(13) = 2 because (13 o)^2 = (6:1 o)^2 = 1.
a(14) = 0 because (14 o)^k = (1:1 4:1 o)^k = 14, for all positive k.
a(15) = 2 because (15 o)^2 = (2:1 3:1 o)^2 = 1.
a(16) = 2 because (16 o)^2 = (1:4 o)^2 = 1.
a(17) = 2 because (17 o)^2 = (7:1 o)^2 = 1.
a(18) = 0 because (18 o)^k = (1:1 2:2 o)^k = 18, for all positive k.
a(19) = 2 because (19 o)^2 = (8:1 o)^2 = 1.
a(20) = 3 because (20 o)^3 = (1:2 3:1 o)^3 = 1.

a(21) = 2 because (21 o)^2 = (2:1 4:1 o)^2 = 1.
a(22) = 0 because (22 o)^k = (1:1 5:1 o)^k = 22, for all positive k.
a(23) = 2 because (23 o)^2 = (9:1 o)^2 = 1.
a(24) = 3 because (24 o)^3 = (1:3 2:1 o)^3 = 1.
a(25) = 2 because (25 o)^2 = (3:2 o)^2 = 1.
a(26) = 0 because (26 o)^k = (1:1 6:1 o)^k = 26, for all positive k.
a(27) = 2 because (27 o)^2 = (2:3 o)^2 = 1.
a(28) = 3 because (28 o)^3 = (1:2 4:1 o)^3 = 1.
a(29) = 2 because (29 o)^2 = (10:1 o)^2 = 1.
a(30) = 0 because (30 o)^k = (1:1 2:1 3:1 o)^k = 30, for all positive k.

a(31) = 2 because (31 o)^k = (11:1 o)^2 = 1.
a(32) = 2 because (32 o)^2 = (1:5 o)^2 = 1.
a(33) = 2 because (33 o)^2 = (2:1 5:1 o)^2 = 1.
a(34) = 0 because (34 o)^k = (1:1 7:1 o)^k = 34, for all positive k.
a(35) = 2 because (35 o)^2 = (3:1 4:1 o)^2 = 1.
a(36) = 0 because (36 o)^k = (1:2 2:2 o)^k = 36, for all positive k.
a(37) = 2 because (37 o)^2 = (12:1 o)^2 = 1.
a(38) = 0 because (38 o)^k = (1:1 8:1 o)^k = 38, for all positive k.
a(39) = 2 because (39 o)^2 = (2:1 6:1 o)^2 = 1.
a(40) = 0 because (40 o)^k in {40, 250}, for all positive k.

a(41) = 2 because (41 o)^2 = (13:1 o)^2 = 1.
a(42) = 0 because (42 o)^k = (1:1 2:1 4:1 o)^k = 42, for all positive k
a(43) = 2 because (43 o)^2 = (14:1 o)^2 = 1.
a(44) = 3 because (44 o)^3 = (1:2 5:1 o)^3 = 1.
a(45) = 0 because (45 o)^k in {45, 9}, for all positive k.
a(46) = 0 because (46 o)^k = (1:1 9:1 o)^k = 46, for all positive k.
a(47) = 2 because (47 o)^2 = (15:1 o)^2 = 1.
a(48) = 3 because (48 o)^3 = (1:4 2:1 o)^3 = 1.
a(49) = 2 because (49 o)^2 = (4:2 o)^2 = 1.
a(50) = 0 because (50 o)^k in {50, 2}, for all positive k.

Detail of calculation for compositional powers of 12:

(12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
(12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12

Detail of calculation for compositional powers of 20:

(20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
(20 o)^3 =     (3:2) o (1:2 3:1) = 1

Detail of calculation for compositional powers of 24:

(24 o)^2 = (1:3 2:1) o (1:3 2:1) = (2:3) = 27
(24 o)^3 =     (2:3) o (1:3 2:1) = 1

Detail of calculation for compositional powers of 28:

(28 o)^2 = (1:2 4:1) o (1:2 4:1) = (4:2) = 49
(28 o)^3 =     (4:2) o (1:2 4:1) = 1

Detail of calculation for compositional powers of 40:

(40 o)^2 = (1:3 3:1) o (1:3 3:1) = (1:1 3:3) = 250
(40 o)^3 = (1:1 3:3) o (1:3 3:1) = (1:3 3:1) =  40

Detail of calculation for compositional powers of 44:

(44 o)^2 = (1:2 5:1) o (1:2 5:1) = (5:2) = 121
(44 o)^3 =     (5:2) o (1:2 5:1) = 1

Detail of calculation for compositional powers of 45:

(45 o)^2 = (2:2 3:1) o (2:2 3:1) = (2:2) = 9
(45 o)^3 =     (2:2) o (2:2 3:1) = (2:2) = 9

Detail of calculation for compositional powers of 48:

(48 o)^2 = (1:4 2:1) o (1:4 2:1) = (2:4) = 81
(48 o)^3 =     (2:4) o (1:4 2:1) = 1

Detail of calculation for compositional powers of 50:

(50 o)^2 = (1:1 3:2) o (1:1 3:2) = (1:1) = 2
(50 o)^3 =     (1:1) o (1:1 3:2) = (1:1) = 2

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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