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 A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers. 20
 0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 32, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 243, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47, 324 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS E.g. we have the following identities: A000040(n) = a(A014580(n)), A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)), A091227(n) = A049084(a(n)), A091247(n) = A066247(a(n)). LINKS Antti Karttunen, Table of n, a(n) for n = 0..8192 A. Karttunen, Scheme-program for computing this sequence. FORMULA a(0)=0, a(1)=1. For n's coding an irreducible polynomial ir_i, that is if n=A014580(i), we have a(n) = A000040(i) and for composite polynomials a(ir_i X ir_j X ...) = p_i * p_j * ..., where p_i = A000040(i) and X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247). Other identities. For all n >= 1, the following holds: A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers. The permutations A091205, A106443, A106445, A106447, A235042 and A245704 have the same property.] From Antti Karttunen, Jun 10 2018: (Start) For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A003961(a(A305422(n))). a(n) = A005940(1+A305418(n)) = A163511(A305428(n)). A046523(a(n)) = A278233(n). (End) PROG (PARI) A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961 A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); A305419(n) = if(n<3, 1, my(k=n-1); while(k>1 && !A091225(k), k--); (k)); A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))), x, 2)); for(i=1, #f~, f[i, 1] = Pol(binary(A305419(f[i, 1])))); fromdigits(Vec(factorback(f))%2, 2); }; A091203(n) = if(n<=1, n, if(!(n%2), 2*A091203(n/2), A003961(A091203(A305422(n))))); \\ Antti Karttunen, Jun 10 2018 CROSSREFS Inverse: A091202. Several variants exist: A235042, A091205, A106443, A106445, A106447. Cf. also A000005, A000040, A001221, A001222, A004247, A008683, A010051, A014580, A048720, A049084, A066247, A091219, A091220, A091221, A091222, A091225, A091227, A091247, A234741, A234742, A245703, A245704, A278233. Cf. also A302024, A302026, A305418, A305428 for other similar permutations. Sequence in context: A260435 A255554 A260741 * A106445 A106443 A091205 Adjacent sequences:  A091200 A091201 A091202 * A091204 A091205 A091206 KEYWORD nonn AUTHOR Antti Karttunen, Jan 03 2004 STATUS approved

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Last modified September 25 20:01 EDT 2020. Contains 337344 sequences. (Running on oeis4.)