

A305422


GF(2)[X] factorization prime shift towards smaller terms.


9



1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 7, 2, 11, 3, 8, 1, 16, 6, 13, 4, 5, 7, 22, 2, 19, 11, 12, 3, 14, 8, 25, 1, 50, 16, 29, 6, 31, 13, 28, 4, 37, 5, 38, 7, 24, 22, 41, 2, 9, 19, 32, 11, 26, 12, 47, 3, 44, 14, 55, 8, 59, 25, 10, 1, 20, 50, 61, 16, 21, 29, 118, 6, 67, 31, 88, 13, 110, 28, 53, 4, 69, 37, 18, 5, 64, 38, 73, 7, 94, 24, 87, 22, 43, 41, 52, 2, 91
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having an unique factorization as f(i) x f(j) x ... x f(k), with 1 <= i <= j <= ... <= k, a(n) = f(i1) x f(j1) x ... x f(k1), where f(0) = 1, and f(n) = A014580(n) for n >= 1.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]


FORMULA

For all n >= 1:
a(A305421(n)) = n.
a(A001317(n)) = A000079(n).
A007814(a(n)) = A268389(n).


PROG

(PARI)
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3, 1, my(k=n1); 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); };


CROSSREFS

Cf. A000079 (positions of ones), A014580, A091225, A268389, A305419, A305421, A305424 (odd bisection), A305425.
Cf. also A064989, A300840.
Sequence in context: A033317 A183200 A326732 * A007733 A128520 A269370
Adjacent sequences: A305419 A305420 A305421 * A305423 A305424 A305425


KEYWORD

nonn


AUTHOR

Antti Karttunen, Jun 07 2018


STATUS

approved



