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%I #24 Aug 06 2015 07:38:59
%S 1361,3721,8073,40257,64125,344925,1121373,4127085,47053305,89025909,
%T 256718241,864417085,2339944761,7793372565,10483463769,15540712857,
%U 19217417625,51731153357,315005744053,731886242745,3047881618969,19546038155241,55232813508469,389828042124021,1225948485247905,17008166929275225
%N Iterates of A234742, starting from value a(0) = 1361, with a(1) = A234742(a(0)), a(2) = A234742(a(1)), etc.
%C 1361 is the first term of A091209 that doesn't reach a fixed point at least for the first 2000 iterations of A234742. Cf. also A260716.
%C Note that 1361 = A048720(61,61).
%H Antti Karttunen, <a href="/A260441/b260441.txt">Table of n, a(n) for n = 0..100</a>
%F a(0) = 1361; for n >= 1, a(n) = A234742(a(n-1)).
%e 61 ("111101" in binary) = A014580(14), i.e., it encodes the fourteenth polynomial with coefficients 0 or 1 that is irreducible over GF(2), namely x^5 + x^4 + x^3 + x^2 + 1. When we multiply that polynomial by itself (in ring GF(2)[X]), we get x^10 + x^8 + x^6 + x^4 + 1, encoded by 1361 with binary representation "10101010001" [1361 = A048720(61,61)]. This is used as the initial value a(0) of this sequence. The next term is obtained by multiplying these two factors 61 and 61 as ordinary integers, which gives a(1) = 61*61 = 3721.
%e 3721 ("111010001001" in binary) in turn encodes polynomial x^11 + x^10 + x^9 + x^7 + x^3 + 1 which factorizes in ring GF(2)[X] as (x + 1)(x + 1)(x + 1)(x^8 + x^5 + x^3 + x + 1). Polynomial (x + 1) is encoded by 3 ("11" in binary) and (x^8 + x^5 + x^3 + x + 1) by 299 ("100101011" in binary). Multiplying 3*3*3*299 in ordinary way gives the next term of the sequence, a(2) = 8073.
%o (PARI)
%o allocatemem((2^30));
%o A234742(n) = factorback(subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2)); \\ After _M. F. Hasler_'s Feb 18 2014 code.
%o 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 iterates_of_A234742(1361, "b260441.txt")
%o (Scheme, with memoizing macro definec)
%o (definec (A260441 n) (if (zero? n) 1361 (A234742 (A260441 (- n 1)))))
%Y Cf. A014580, A048720, A234742, A260712, A260713, A260716.
%Y Cf. A260720 (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 the next step).
%Y Subsequence of A016813.
%Y Cf. also A244323, A260729, A260735 for iterations starting from other values.
%K nonn
%O 0,1
%A _Antti Karttunen_, Aug 04 2015
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