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Integers i such that 19*i = 55 X i.
4

%I #17 Jan 04 2022 09:23:06

%S 0,7,14,28,31,56,62,63,112,119,124,126,127,224,238,248,252,254,255,

%T 448,455,476,496,504,508,510,511,896,910,952,992,1008,1016,1020,1022,

%U 1023,1792,1799,1820,1823,1904,1911,1984,1991,2016,2032,2040,2044

%N Integers i such that 19*i = 55 X i.

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

%C From _Robert Israel_, Apr 08 2018: (Start)

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

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

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

%H Robert Israel, <a href="/A115874/b115874.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>

%p X:= proc(a,b) local A,B,C;

%p A:= convert(a,base,2);

%p B:= convert(b,base,2);

%p C:= expand(add(A[i]*x^(i-1),i=1..nops(A))*add(B[i]*x^(i-1),i=1..nops(B))) mod 2;

%p eval(C,x=2)

%p end proc:

%p select(t -> 19*t = X(55,t), [$0..10^4]); # _Robert Israel_, Apr 08 2018

%t X[a_, b_] := Module[{A, B, C},

%t A = Reverse@IntegerDigits[a, 2];

%t B = Reverse@IntegerDigits[b, 2];

%t C = Expand[

%t Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*

%t Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];

%t PolynomialMod[C, 2] /. x -> 2];

%t Select[Range[0, 10^4], 19*# == 55~X~#&] (* _Jean-François Alcover_, Jan 04 2022, after _Robert Israel_ *)

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

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 07 2006

%E Offset corrected by _Robert Israel_, Apr 08 2018