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A260426 a(1) = 1, a(A206074(n)) = A014580(a(n)), a(A205783(1+n)) = A091242(a(n)), where A014580 [respectively A091242] give binary codes for irreducible [resp. reducible] polynomials over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q. 8

%I

%S 1,2,3,4,7,5,11,6,8,12,25,9,13,17,10,14,47,18,19,34,15,20,31,24,55,16,

%T 21,62,137,26,37,27,45,22,28,42,59,33,71,23,87,29,41,79,166,35,61,49,

%U 36,58,30,38,319,54,91,76,44,89,97,32,203,108,39,53,99,200,67,46,103,78,185,64,131,48,75,40,379,50,73,373,109,70,433,113,95,57,1123,111,143,121

%N a(1) = 1, a(A206074(n)) = A014580(a(n)), a(A205783(1+n)) = A091242(a(n)), where A014580 [respectively A091242] give binary codes for irreducible [resp. reducible] polynomials over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

%C Each term of A260427 resides in a separate infinite cycle. This follows because any polynomial with (coefficients 0 or 1) that is irreducible over GF(2) is also irreducible over Q, in other words, A014580 is a subset of A206074. [See _Thomas Ordowski_'s Feb 21 2014 comment in A014580] and thus any term of A091242 in A206074 is trapped into a trajectory containing only terms of A014580.

%H Antti Karttunen, <a href="/A260426/b260426.txt">Table of n, a(n) for n = 1..7968</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A014580(a(A255574(n))), otherwise [when n is in A205783], a(n) = A091242(a(A255572(n))).

%F As a composition of related permutations:

%F a(n) = A246202(A260421(n)).

%F a(n) = A245703(A260424(n)).

%o (PARI)

%o allocatemem(234567890);

%o vecsize = (2^24)-4;

%o uplim = 2^25;

%o v014580 = vector(vecsize); A014580 = n -> v014580[n];

%o v091242 = vector(vecsize); A091242 = n -> v091242[n];

%o v255574 = vector(vecsize); A255574 = n -> v255574[n];

%o A255572 = n -> (n - A255574(n) - 1);

%o isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_

%o isA206074(n) = polisirreducible(Pol(binary(n)));

%o A257000 = n -> isA206074(n);

%o v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), if(!(n%65536),print1(n,", ")); v255574[n] = v255574[n-1]+A257000(n); if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++); print(n);

%o A260426(n) = if(1==n, 1, if(isA206074(n), A014580(A260426(A255574(n))), A091242(A260426(A255572(n)))));

%o for(n=1, 7968, write("b260426.txt", n, " ", A260426(n)));

%Y Inverse: A260425.

%Y Related permutations: A246202, A245703, A260421, A260424.

%Y Cf. A014580, A091242, A205783, A206074, A257000, A255572, A255574, A260427.

%Y Differs from A245703 for the first time at n=25, where a(25)=55, while A245703(25)=16.

%K nonn,look

%O 1,2

%A _Antti Karttunen_, Jul 26 2015

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Last modified January 23 01:15 EST 2020. Contains 331166 sequences. (Running on oeis4.)