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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(i-1) x f(j-1) x ... x f(k-1), 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=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); };

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

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)