A108352 a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371 and (n o)^k = n o ... o n, with k occurrences of n.

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, 2, 3, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 3, 3, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 0, 2, 2, 0, 2, 3, 2

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Jon Awbrey, Riffs and Rotes

Jon Awbrey, Primal Code Characteristic, n = 1 to 3000 (Note: values given here differ at n = 718, 746, 1156, 1449, 1734 and 1804 from those computed in b-file). - Antti Karttunen, Nov 23 2019

Formula

a(A065091(n)) = 2 for all n, a(A001747(n)) = 0 for all n, except n=2, and a(A046315(n)) = 2 for n > 1. - Antti Karttunen, Nov 20 2019

Example

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.
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.
From Antti Karttunen, Nov 20 2019: (Start)
For n=718, because 718 = prime(1)^1 * prime(72)^1, its partial function primal code is (1:1 72:1), which, when composed with itself stays same (that is, A106177(718,718) = 718), thus, as 1 is never reached, a(718) = 0, like is true for all even nonsquare semiprimes.
For n=1804, as 1804 = prime(1)^2 * prime(5)^1 * prime(13)^1, its primal code is (1:2 5:1 13:1), which, when composed with itself yields 203401 = prime(5)^2 * prime(13)^2, i.e., primal code (5:2 13:2), which when composed with (1:2 5:1 13:1) yields 1, which happened on the second iteration, thus a(1804) = 2+1 = 3.
(End)

Prog

(PARI)
A106177sq(n, k) = { my(f = factor(k)); prod(i=1, #f~, f[i, 1]^valuation(n, prime(f[i, 2]))); }; \\ As in A106177.
A108352(n) = { my(orgn=n, xs=Set([]), k=1); while(n>1, if(vecsearch(xs, n), return(0)); xs = setunion([n], xs); n = A106177sq(n, orgn); k++); (k); }; \\ Antti Karttunen, Nov 20 2019

Crossrefs

Cf. A001747, A046315, A055231, A061396, A062504, A062537, A062860, A065091, A106177, A106178.

Cf. A108353, A108370, A108371, A108372, A108373, A108374, A111801.

Sequence in context: A138319 A217864 A002100 * A346149 A215883 A277024

Adjacent sequences: A108349 A108350 A108351 * A108353 A108354 A108355

Keyword

nonn

Author

Jon Awbrey, May 31 2005, revised Jun 01 2005

Extensions

Links and cross-references added, Aug 19 2005

Term a(63) corrected and five more terms added (up to a(105)) by Antti Karttunen, Nov 20 2019

Status

approved