A391575 - OEIS

%I #24 Dec 18 2025 10:16:33

%S 3,6,11,12,13,15,19,22,24,25,26,30,35,38,43,44,45,47,48,49,50,51,52,

%T 53,55,59,60,61,63,67,70,75,76,77,79,83,86,88,89,90,94,96,97,98,99,

%U 100,101,102,104,105,106,110,118,120,121,122,126,131,134,139,140,141,143,147,150,152,153,154,158,163,166,171,172

%N Numbers k such that 3*k = A048720(3,m) for some m > k, where A048720 is carryless base-2 multiplication.

%C Numbers k whose binary expansion contains 2 adjacent 1's (i.e., is not a Fibbinary number, A003714) and for which 3*k = A048724(m) for some m, where m is then by necessity > k.

%C If n is present, then 2*n is also present, and vice versa.

%H Antti Karttunen, <a href="/A391575/b391575.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>.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>.

%e 3, which in binary is A007088(3) = "11" (with two adjacent 1-bits), and 3*3 = 9, with A007088(9) = "1001", the same bit-pattern can be obtained by xoring "111" (= A007088(7)) with itself shifted once left, that is, as 9 = A048724(7), thus 3*3 = A048720(3,7), and because 7 > 3, therefore 3 is included in this sequence.

%e 51, which in binary is A007088(51) = "110011" (with two adjacent 1-bits), and 3*51 = 153, with A007088(153) = "10011001", the same bit-pattern can be obtained by xoring "1110111" [= A007088(119)] with itself shifted once left, that is, as 153 = A048724(119), thus 3*51 = A048720(3,119), and because 119 > 51, therefore 51 is included in this sequence.

%o (PARI) is_A391575(n) = ((bitand(n, n>>1)>0) && 0==lift((Pol(binary(3*n))*Mod(1, 2)) % (Pol(binary(3))*Mod(1, 2))));

%o (PARI)

%o A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);

%o is_A391575(n) = (0==lift((Pol(binary(3*n))*Mod(1, 2)) % (Pol(binary(3))*Mod(1, 2))) && (A048720(3,n) != 3*n));

%o (PARI)

%o A280500_sq(a, b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); if(0!=lift(Pa % Pb), 0, fromdigits(Vec(lift(Pa/Pb)), 2)); };

%o is_A391575(k) = { my(m=A280500_sq(3*k, 3)); (m>k); };

%Y Cf. A007088, A048720, A048724, A280500, A391570.

%Y Intersection of A004780 and A391574. Setwise difference A391574 \ A003714.

%Y First differs from A022155 at a(22) = 51, as that value is missing from A022155.

%Y Cf. also A391577.

%K nonn,base

%O 1,1

%A _Antti Karttunen_, Dec 17 2025