A305422 GF(2)[X] factorization prime shift towards smaller terms.

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

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 a 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