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 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. 21
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 20 03:21 EDT 2024. Contains 373512 sequences. (Running on oeis4.)