A391575 Numbers k such that 3*k = A048720(3,m) for some m > k, where A048720 is carryless base-2 multiplication.

3, 6, 11, 12, 13, 15, 19, 22, 24, 25, 26, 30, 35, 38, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 59, 60, 61, 63, 67, 70, 75, 76, 77, 79, 83, 86, 88, 89, 90, 94, 96, 97, 98, 99, 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

Offset

1,1

Comments

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.

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences defined by congruent products between domains N and GF(2)[X].

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to carryless arithmetic.

Example

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.
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.

Prog

(PARI) is_A391575(n) = ((bitand(n, n>>1)>0) && 0==lift((Pol(binary(3*n))*Mod(1, 2)) % (Pol(binary(3))*Mod(1, 2))));
(PARI)
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
is_A391575(n) = (0==lift((Pol(binary(3*n))*Mod(1, 2)) % (Pol(binary(3))*Mod(1, 2))) && (A048720(3, n) != 3*n));
(PARI)
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)); };
is_A391575(k) = { my(m=A280500_sq(3*k, 3)); (m>k); };

Crossrefs

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

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

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

Cf. also A391577.

Sequence in context: A028744 A028775 A223910 * A022155 A066157 A073159

Adjacent sequences: A391572 A391573 A391574 * A391576 A391577 A391578

Keyword

nonn,base

Author

Antti Karttunen, Dec 17 2025

Status

approved