Riffs & Rotes

[Jon Awbrey]

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R&R.  Note 8

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SeqFans,

Have just submitted the following sequence,
which may suggest a genre of related items:

A108352 [pending]

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

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.

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

| N.J.A. Sloane,
|'On-Line Encyclopedia of Integer Sequences',
| In: http://www.research.att.com/~njas/sequences/
| Cf. http://www.research.att.com/projects/OEIS?Anum=A055231
| Cf. http://www.research.att.com/projects/OEIS?Anum=A061396
| Cf. http://www.research.att.com/projects/OEIS?Anum=A062504
| Cf. http://www.research.att.com/projects/OEIS?Anum=A062537
| Cf. http://www.research.att.com/projects/OEIS?Anum=A062860
| Cf. http://www.research.att.com/projects/OEIS?Anum=A106177
| Cf. http://www.research.att.com/projects/OEIS?Anum=A106178

Anyway, I hope there aren't too many errors in this.

Jon Awbrey

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