cp's OEIS Frontend

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

%I A268390 #62 Feb 16 2025 08:33:30
%S A268390 1,6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,62,65,69,74,
%T A268390 77,82,85,86,87,91,93,94,95,100,106,111,115,118,119,122,123,129,133,
%U A268390 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
%N A268390 Products of an even number of distinct primes and the square of a number in the sequence (including 1).
%C A268390 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.
%C A268390 From _Peter Munn_, Sep 14 2019 and Dec 01 2019: (Start)
%C A268390 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.
%C A268390 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.
%C A268390 (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.
%C A268390 (End)
%C A268390 From _Peter Munn_, Dec 01 2021: (Start)
%C A268390 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.
%C A268390 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.
%C A268390 (End)
%H A268390 Antti Karttunen, <a href="/A268390/b268390.txt">Table of n, a(n) for n = 1..10000</a>
%H A268390 OEIS Wiki, <a href="/wiki/Ideals">Ideal</a>.
%H A268390 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Closure.html">Closure</a>, <a href="https://mathworld.wolfram.com/Group.html">Group</a>, <a href="https://mathworld.wolfram.com/LeftTransversal.html">Left Transversal</a>, <a href="https://mathworld.wolfram.com/RightTransversal.html">Right Transversal</a>, <a href="https://mathworld.wolfram.com/SquarePart.html">Square part</a>, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree part</a>.
%F A268390 From _Peter Munn_, Oct 30 2019: (Start)
%F A268390 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}.
%F A268390 For n >= 1, k >= 0, prime p, A268387(A059897(p^k, a(n))) = k.
%F A268390 (End)
%F A268390 From _Peter Munn_, Nov 24 2021: (Start)
%F A268390 {a(n) : n >= 1} = {A000188(a(n)) : n >= 1}.
%F A268390 {a(n) : n >= 1} = {A225546(A262675(n)) : n >= 1}.
%F A268390 {A059897(a(n), A262675(m)) : n >= 1, m >= 1} = {A000379(k) : k >= 1}.
%F A268390 (End)
%e A268390 1 has an empty factorization, and as XOR of an empty set is zero, 1 is included.
%e A268390 6 = 2^1 * 3^1 and as XOR(1,1) = 0, 6 is included.
%e A268390 30 = 2^1 * 3^1 * 5^1 is NOT included, as XOR(1,1,1) = 1.
%e A268390 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.
%e A268390 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
%t A268390 Select[Range[200], # == 1 || BitXor @@ Last /@ FactorInteger[#] == 0 &] (* _Amiram Eldar_, Nov 27 2020 *)
%o A268390 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A268390 (define A268390 (ZERO-POS 1 1 A268387))
%Y A268390 Positions of 0's in A268387, cf. A374595 (positions of 1's).
%Y A268390 Cf. A000188, A003987, A048833 (counts prime signatures that are represented), A059897, A329329.
%Y A268390 Subsequences: A006881 (semiprime terms), A030229 (squarefree terms), A238748 (differs first by missing a(115) = 360 and lists more subsequences).
%Y A268390 Subsequences for prime signatures not within A238748: A163569, A190111, A190468.
%Y A268390 Subsequence of A000379, A028260. Differs from their intersection, A374472, by omitting 64, 144, 324 etc.
%Y A268390 Related to A262675 via A225546.
%Y A268390 Ordered odd bisection of A334205.
%K A268390 nonn,base
%O A268390 1,2
%A A268390 _Antti Karttunen_, Feb 05 2016
%E A268390 New name from _Peter Munn_, Jul 15 2024