A260720 -id:A260720 - cp's OEIS frontend

A260441 Iterates of A234742, starting from value a(0) = 1361, with a(1) = A234742(a(0)), a(2) = A234742(a(1)), etc.

1361, 3721, 8073, 40257, 64125, 344925, 1121373, 4127085, 47053305, 89025909, 256718241, 864417085, 2339944761, 7793372565, 10483463769, 15540712857, 19217417625, 51731153357, 315005744053, 731886242745, 3047881618969, 19546038155241, 55232813508469, 389828042124021, 1225948485247905, 17008166929275225

Offset: 0

Antti Karttunen, Aug 04 2015

nonn

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.

Note that 1361 = A048720(61,61).

			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.
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.

Antti Karttunen, Table of n, a(n) for n = 0..100

Cf. A014580, A048720, A234742, A260712, A260713, A260716.
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).
Subsequence of A016813.
Cf. also A244323, A260729, A260735 for iterations starting from other values.

allocatemem((2^30));
A234742(n) = factorback(subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2)); \\ After M. F. Hasler's Feb 18 2014 code.
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.
iterates_of_A234742(1361, "b260441.txt")

;; With memoizing macro definec.
(definec (A260441 n) (if (zero? n) 1361 (A234742 (A260441 (- n 1)))))

a(0) = 1361; for n >= 1, a(n) = A234742(a(n-1)).

A260719 a(n) = A091222(A260735(n)): number of irreducible factors (in ring GF(2)[X]) of the binary encoded polynomial obtained after the n-th iteration of A234742, when starting with the initial value 455.

Original entry on oeis.org

5, 5, 3, 4, 3, 5, 9, 2, 7, 2, 6, 6, 2, 2, 2, 3, 8, 8, 6, 5, 5, 7, 6, 4, 5, 6, 2, 7, 6, 4, 5, 4, 5, 4, 5, 9, 4, 10, 3, 4, 7, 4, 4, 3, 4, 3, 5, 8, 6, 4, 7, 5, 3, 7, 3, 3, 3, 3, 3, 7, 3, 5, 6, 6, 9, 4, 9, 3, 5, 6, 3, 4, 5, 7, 7, 4, 5, 2, 10, 11, 6, 6, 7, 13, 4, 6, 5, 10, 6, 4, 7, 4, 10, 8, 3, 7, 7, 4, 5, 5, 2, 4, 8, 3, 4, 3, 7, 4, 6, 3, 15, 3, 4, 7, 6, 6, 6, 5, 5, 8, 4

Offset: 0

Antti Karttunen, Aug 04 2015

nonn

Records occur in positions 0, 6, 37, 79, 83, 110, 329, 554, 1019, 1318, 2027, and they are 5, 9, 10, 11, 13, 15, 16, 17, 20, 21, 23.
First 2's occur at positions 7, 9, 12, 13, 14, 26, 77, 100, 127, 158, 161, 173, 183, 193, 201, 208, 442, 447, 528, 544, 642, 706, 1033, 1089, 1222, 1831.
Note that if this sequence ever obtains value 1, then the rest of terms are also 1's, as then A260735 has attained as its constant value one of the terms of A091214 (which is a subsequence of A235035, the fixed points of A234742).

			See example in A260735. This sequence gives the number of those irreducible factors (in ring GF(2)[X], not necessarily all primes in Z) that are multiplied together (in ordinary way) to get the next term of A260735. For example, a(0) = 5 (for 3 * 3 * 7 * 7 * 7), a(1) = 5 (for 3 * 7 * 7 * 13 * 13).

Antti Karttunen, Table of n, a(n) for n = 0..2049

Cf. A091214, A091222, A234742, A235035, A260735, A260720.

allocatemem((2^30));
{my(n=455, fm); for(i=0,2049, fm=factor(Pol(binary(n))*Mod(1, 2)); write("b260719.txt", i, " ", sum(k=1, #fm~, fm[k, 2])); n = factorback(subst(lift(fm),x,2))); };

(define (A260719 n) (A091222 (A260735 n)))

a(n) = A091222(A260735(n)).

cp's OEIS Frontend

A260441 Iterates of A234742, starting from value a(0) = 1361, with a(1) = A234742(a(0)), a(2) = A234742(a(1)), etc.

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