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

%I A115874 #17 Jan 04 2022 09:23:06
%S A115874 0,7,14,28,31,56,62,63,112,119,124,126,127,224,238,248,252,254,255,
%T A115874 448,455,476,496,504,508,510,511,896,910,952,992,1008,1016,1020,1022,
%U A115874 1023,1792,1799,1820,1823,1904,1911,1984,1991,2016,2032,2040,2044
%N A115874 Integers i such that 19*i = 55 X i.
%C A115874 Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
%C A115874 From _Robert Israel_, Apr 08 2018: (Start)
%C A115874 n is in the sequence if and only if 2*n is.
%C A115874 If n is in the sequence, then so is (2^k+1)*n if 2^k > n.
%C A115874 Contains 2^k-1 for k >= 5. (End)
%H A115874 Robert Israel, <a href="/A115874/b115874.txt">Table of n, a(n) for n = 1..10000</a>
%H A115874 <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>
%p A115874 X:= proc(a,b) local A,B,C;
%p A115874 A:= convert(a,base,2);
%p A115874 B:= convert(b,base,2);
%p A115874 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 A115874 eval(C,x=2)
%p A115874 end proc:
%p A115874 select(t -> 19*t = X(55,t), [$0..10^4]); # _Robert Israel_, Apr 08 2018
%t A115874 X[a_, b_] := Module[{A, B, C},
%t A115874      A = Reverse@IntegerDigits[a, 2];
%t A115874      B = Reverse@IntegerDigits[b, 2];
%t A115874      C = Expand[
%t A115874         Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*
%t A115874         Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];
%t A115874      PolynomialMod[C, 2] /. x -> 2];
%t A115874 Select[Range[0, 10^4], 19*# == 55~X~#&] (* _Jean-François Alcover_, Jan 04 2022, after _Robert Israel_ *)
%Y A115874 Row 19 of A115872. Superset of A115876? A115875 shows this sequence in binary.
%K A115874 nonn
%O A115874 1,2
%A A115874 _Antti Karttunen_, Feb 07 2006
%E A115874 Offset corrected by _Robert Israel_, Apr 08 2018