A268390 Products of an even number of distinct primes and the square of a number in the sequence (including 1).

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 196, 201, 202, 203, 205, 206, 209, 210

Offset: 1

Author

Antti Karttunen, Feb 05 2016

Keywords

• nonn
• base

Comments

Old name: 'Positions of zeros in A268387: numbers n such that when the exponents e_1 .. e_k in their prime factorization n = p_1^e_1 * ... * p_k^e_k are bitwise-xored together, the result is zero.

From Peter Munn, Sep 14 2019 and Dec 01 2019: (Start)

When trailing zeros are removed from the terms written in base p, for any prime p, every positive integer not divisible by p appears exactly once. This is the lexicographically earliest sequence with this property.

The closure of A238748 with respect to the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), the sequence thereby forms a subgroup, denoted H, of the positive integers under A059897(.,.). H is a subgroup of A000379.

(The symbol ^ can take on a meaning in relation to a group operation. However, in this comment ^ denotes the power operator for standard integer multiplication.) For any prime p, the subgroup {p^k : k >= 0} and H are each a (left and right) transversal of the other. For k >= 0 and primes p_1 and p_2, the cosets (p_1^k)H and (p_2^k)H are the same.

(End)

From Peter Munn, Dec 01 2021: (Start)

If we take the square root of the square terms we reproduce the sequence itself. The set of all products of a square term and a squarefree term is the sequence as a set.

The terms are the elements of the ideal generated by {6} in the ring defined in A329329. Similarly, the ideal generated by {8} gives A262675. 6 and 8 are images of each other under A225546(.), which is an automorphism of the ring. So this sequence and A262675, as sets, are images of each other under A225546(.). The elements of the ideal generated by {6,8} form the notable set A000379.

(End)

Examples

			1 has an empty factorization, and as XOR of an empty set is zero, 1 is included.
6 = 2^1 * 3^1 and as XOR(1,1) = 0, 6 is included.
30 = 2^1 * 3^1 * 5^1 is NOT included, as XOR(1,1,1) = 1.
360 = 2^3 * 3^2 * 5^1 is included, as the bitwise-XOR of exponents 3, 2 and 1 ("11", "10" and "01" in binary) results zero.
10, 15, 36 and 216 are in A238748. 360 = A059897(10, 36) = A059897(15, 216) and 540 = A059897(15, 36) = A059897(10, 216). So 360 and 540 are in the closure of A238748 under A059897(.,.), so in this sequence although absent from A238748. - _Peter Munn_, Oct 30 2019
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• OEIS Wiki, Ideal.
• Eric Weisstein's World of Mathematics, Closure, Group, Left Transversal, Right Transversal, Square part, Squarefree part.

Crossrefs

Positions of 0's in A268387, cf. A374595 (positions of 1's).

Cf. A000188, A003987, A048833 (counts prime signatures that are represented), A059897, A329329.

Subsequences: A006881 (semiprime terms), A030229 (squarefree terms), A238748 (differs first by missing a(115) = 360 and lists more subsequences).

Subsequences for prime signatures not within A238748: A163569, A190111, A190468.

Subsequence of A000379, A028260. Differs from their intersection, A374472, by omitting 64, 144, 324 etc.

Related to A262675 via A225546.

Ordered odd bisection of A334205.

Programs

Mathematica

Select[Range[200], # == 1 || BitXor @@ Last /@ FactorInteger[#] == 0 &] (* Amiram Eldar, Nov 27 2020 *)

Formula

From Peter Munn, Oct 30 2019: (Start)

For k >= 0, prime p_1, prime p_2, {m : m = A059897(p_1^k, a(n)), n >= 1} = {m : m = A059897(p_2^k, a(n)), n >= 1}.

For n >= 1, k >= 0, prime p, A268387(A059897(p^k, a(n))) = k.

(End)

From Peter Munn, Nov 24 2021: (Start)

{a(n) : n >= 1} = {A000188(a(n)) : n >= 1}.

{a(n) : n >= 1} = {A225546(A262675(n)) : n >= 1}.

{A059897(a(n), A262675(m)) : n >= 1, m >= 1} = {A000379(k) : k >= 1}.

(End)

Extensions

New name from Peter Munn, Jul 15 2024