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].

10, 12, 18, 20, 24, 40, 56, 88, 104, 116, 136, 184, 196, 224, 312, 368, 428, 464, 520, 528, 650, 672, 760, 884, 992, 1472, 1504, 1888, 1952, 2528, 3424, 3724, 4832, 5312, 6464, 7136, 9112, 11096, 11288, 11744, 13216, 15352, 15376, 15872, 15968, 16256, 17816, 17964, 22616, 24448, 26728, 28544, 29296, 30592, 30656

Offset

1,1

Comments

Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.

Links

Antti Karttunen, Table of n, a(n) for n = 1..484; all terms <= 2^33

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to polynomials in ring GF(2)[X].

Index entries for sequences related to sigma(n).

Formula

{k such that k = A048720(A318467(k), x) for some x > 0}.

{k not in A000396 such that A280500(k, A318467(k)) > 0}.

Example

196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.

Prog

(PARI)
divides_in_GF2X(a, b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };

Crossrefs

Cf. A000203, A000396, A003987, A048720, A280500, A318467.

Cf. A379234 (subsequence).

Cf. also A097498 (= A153501 U A271816).

Sequence in context: A327709 A129120 A134516 * A144363 A051247 A242335

Adjacent sequences: A379233 A379234 A379235 * A379237 A379238 A379239

Keyword

nonn

Author

Antti Karttunen, Jan 05 2025

Status

approved