A115874 Integers i such that 19*i = 55 X i.

0, 7, 14, 28, 31, 56, 62, 63, 112, 119, 124, 126, 127, 224, 238, 248, 252, 254, 255, 448, 455, 476, 496, 504, 508, 510, 511, 896, 910, 952, 992, 1008, 1016, 1020, 1022, 1023, 1792, 1799, 1820, 1823, 1904, 1911, 1984, 1991, 2016, 2032, 2040, 2044

Offset: 1

Author

Antti Karttunen, Feb 07 2006

Keywords

• nonn

Comments

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

From Robert Israel, Apr 08 2018: (Start)

n is in the sequence if and only if 2*n is.

If n is in the sequence, then so is (2^k+1)*n if 2^k > n.

Contains 2^k-1 for k >= 5. (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Crossrefs

Row 19 of A115872. Superset of A115876? A115875 shows this sequence in binary.

Programs

Maple

X:= proc(a,b) local A,B,C;
A:= convert(a,base,2);
B:= convert(b,base,2);
C:= expand(add(A[i]*x^(i-1),i=1..nops(A))*add(B[i]*x^(i-1),i=1..nops(B))) mod 2;
eval(C,x=2)
end proc:
select(t -> 19*t = X(55,t), [$0..10^4]); # Robert Israel, Apr 08 2018

Mathematica

X[a_, b_] := Module[{A, B, C},
     A = Reverse@IntegerDigits[a, 2];
     B = Reverse@IntegerDigits[b, 2];
     C = Expand[
        Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*
        Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];
     PolynomialMod[C, 2] /. x -> 2];
Select[Range[0, 10^4], 19*# == 55~X~#&] (* Jean-François Alcover, Jan 04 2022, after Robert Israel *)

Extensions

Offset corrected by Robert Israel, Apr 08 2018