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A260735 Iterates of A234742, starting from value a(0) = 455, with a(1) = A234742(a(0)), a(2) = A234742(a(1)), etc.

%I A260735 #22 Jan 25 2025 13:06:26
%S A260735 455,3087,24843,72975,332563,602919,5893875,221402727,322063831,
%T A260735 5853742587,10696444275,75642464331,749833439355,1724537517955,
%U A260735 2295761459035,4498164915283,9436077956619,369311889576231,10610033249983167,135786986032294135,460149860040811083,2879918014301480295,63102417694969716063,339029616686070752991
%N A260735 Iterates of A234742, starting from value a(0) = 455, with a(1) = A234742(a(0)), a(2) = A234742(a(1)), etc.
%C A260735 455 is the first term of A236844 that doesn't settle to a fixed point at least for the first 2000 iterations of A234742. Cf. also A260713.
%H A260735 Antti Karttunen, <a href="/A260735/b260735.txt">Table of n, a(n) for n = 0..100</a>
%F A260735 a(0) = 455; for n >= 1, a(n) = A234742(a(n-1)).
%e A260735 The initial value a(0) = 455 ("111000111" in binary) encodes polynomial (with coefficients 0 or 1) x^8 + x^7 + x^6 + x^2 + x + 1, which in ring GF(2)[X] factorizes as (x + 1)(x + 1)(x^2 + x + 1)(x^2 + x + 1)(x^2 + x + 1). (x+1) is encoded by 3 ("11" in binary) and (x^2 + x + 1) by 7 ("111" in binary). Multiplying 3*3*7*7*7 yields the next term of the sequence, thus a(1) = 3087.
%e A260735 3087 ("110000001111" in binary) in turn encodes polynomial x^11 + x^10 + x^3 + x^2 + x + 1 which factorizes as (x + 1)(x^2 + x + 1)(x^2 + x + 1)(x^3 + x^2 + 1)(x^3 + x^2 + 1). Polynomial (x^3 + x^2 + 1) is encoded by 13, as 13 is "1101" in binary. Multiplying 3*7*7*13*13 yields the next term of the sequence, a(2) = 24843.
%o A260735 (PARI)
%o A260735 allocatemem((2^30));
%o A260735 A234742(n) = factorback(subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2)); \\ After _M. F. Hasler_'s Feb 18 2014 code.
%o A260735 iterates_of_A234742(start, filename) = {my(n=start, prev=-1, prevprev=-1, i=0); until((n==prevprev), write(filename, i, " ", n); prevprev = prev; prev = n; n = A234742(n); i++)} \\ Computes b-file up to the second occurrence of the fixed point or until the user presses Ctrl-C.
%o A260735 iterates_of_A234742(455, "b260735.txt")
%o A260735 (Scheme)
%o A260735 ;; With memoizing macro definec.
%o A260735 (definec (A260735 n) (if (zero? n) 455 (A234742 (A260735 (- n 1)))))
%Y A260735 Cf. A234742, A260712, A260713.
%Y A260735 Cf. A260719  (for each term, gives the number of irreducible factors in ring GF(2)[X] for the corresponding encoded polynomial, equal to how many numbers are multiplied together at each step).
%Y A260735 Subsequence of A004767.
%Y A260735 Cf. also A244323, A260729, A260441 for iterations starting from other values.
%K A260735 nonn
%O A260735 0,1
%A A260735 _Antti Karttunen_, Aug 04 2015