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

Author

Antti Karttunen, Jun 07 2018

Keywords

• nonn

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]

Crossrefs

Cf. A000079 (positions of ones), A014580, A091225, A268389, A305419, A305421, A305424 (odd bisection), A305425.

Cf. also A064989, A300840.

Programs

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); };

Formula

For all n >= 1:

a(A305421(n)) = n.

a(A001317(n)) = A000079(n).

A007814(a(n)) = A268389(n).