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A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

%I A379236 #19 Mar 31 2025 07:00:08
%S A379236 10,12,18,20,24,40,56,88,104,116,136,184,196,224,312,368,428,464,520,
%T A379236 528,650,672,760,884,992,1472,1504,1888,1952,2528,3424,3724,4832,5312,
%U A379236 6464,7136,9112,11096,11288,11744,13216,15352,15376,15872,15968,16256,17816,17964,22616,24448,26728,28544,29296,30592,30656
%N A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].
%C A379236 Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.
%H A379236 Antti Karttunen, <a href="/A379236/b379236.txt">Table of n, a(n) for n = 1..484; all terms <= 2^33</a>
%H A379236 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A379236 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>.
%H A379236 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379236 {k such that k = A048720(A318467(k), x) for some x > 0}.
%F A379236 {k not in A000396 such that A280500(k, A318467(k)) > 0}.
%e A379236 196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.
%o A379236 (PARI)
%o A379236 divides_in_GF2X(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
%o A379236 is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };
%Y A379236 Cf. A000203, A000396, A003987, A048720, A280500, A318467.
%Y A379236 Cf. A379234 (subsequence).
%Y A379236 Cf. also A097498 (= A153501 U A271816).
%K A379236 nonn
%O A379236 1,1
%A A379236 _Antti Karttunen_, Jan 05 2025