A260719 -id:A260719 - cp's OEIS frontend

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

455, 3087, 24843, 72975, 332563, 602919, 5893875, 221402727, 322063831, 5853742587, 10696444275, 75642464331, 749833439355, 1724537517955, 2295761459035, 4498164915283, 9436077956619, 369311889576231, 10610033249983167, 135786986032294135, 460149860040811083, 2879918014301480295, 63102417694969716063, 339029616686070752991

Offset: 0

Antti Karttunen, Aug 04 2015

nonn

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.

			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.
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.

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

Cf. A234742, A260712, A260713.
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).
Subsequence of A004767.
Cf. also A244323, A260729, A260441 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(455, "b260735.txt")

;; With memoizing macro definec.
(definec (A260735 n) (if (zero? n) 455 (A234742 (A260735 (- n 1)))))

a(0) = 455; for n >= 1, a(n) = A234742(a(n-1)).

A260720 a(n) = A091222(A260441(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 1361.

Original entry on oeis.org

2, 4, 5, 2, 6, 4, 4, 8, 3, 3, 4, 3, 3, 3, 2, 2, 3, 5, 2, 4, 7, 2, 5, 3, 7, 3, 3, 4, 4, 7, 4, 6, 5, 3, 2, 5, 6, 4, 8, 4, 4, 6, 3, 4, 5, 3, 3, 4, 5, 6, 6, 6, 3, 6, 10, 6, 4, 5, 6, 8, 3, 3, 5, 3, 8, 2, 3, 4, 5, 6, 5, 4, 5, 5, 7, 4, 5, 6, 3, 5, 6, 5, 6, 7, 3, 8, 7, 10, 7, 9, 6, 5, 2, 6, 5, 7, 6, 8, 6, 3, 10, 3, 9, 8, 6, 6, 5, 8, 6, 7, 3, 6, 8, 5, 5, 5, 8, 5, 6, 5, 7

Offset: 0

Antti Karttunen, Aug 04 2015

nonn

Records occur in positions 0, 1, 2, 4, 7, 54, 139, 174, 225, 398, 778, and they are 2, 4, 5, 6, 8, 10, 11, 13, 16, 20, 21.
First 2's occur at positions 0, 3, 14, 15, 18, 21, 34, 65, 92, 135, 200, 255, 339, 362, 468, 511, 825, 1042, 1809.
Note that if this sequence ever obtains value 1, then the rest of terms are also 1's, as then A260441 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 A260441. This sequence gives the number of those irreducible factors, counted with multiplicity. For example, a(0) = 2 (for 61 * 61), a(1) = 4 (for 3 * 3 * 3 * 299). Note that irreducibility here refers to irreducibility in ring GF(2)[X], as for example 299 = 13*23 when factored to ordinary primes.

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

Cf. A091214, A091222, A234742, A235035, A260441, A260719.

allocatemem((2^30));
{my(n=1361, fm); for(i=0,2049, fm=factor(Pol(binary(n))*Mod(1, 2)); write("b260720.txt", i, " ", sum(k=1, #fm~, fm[k, 2])); n = factorback(subst(lift(fm),x,2))); };

(define (A260720 n) (A091222 (A260441 n)))

a(n) = A091222(A260441(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula