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
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