cp's OEIS Frontend

A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

%I A370267 #12 Mar 19 2025 18:53:16
%S A370267 1,4,6,7,9,10,15,16,17,22,23,24,25,26,28,31,33,36,38,39,40,41,42,47,
%T A370267 49,54,55,57,58,60,63,64,65,68,70,71,73,74,79,81,86,87,88,89,90,92,95,
%U A370267 96,97,100,102,103,104,105,106,111,112,113,118,119,121,122,124,127,129
%N A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).
%C A370267 Construction by subgroup generation: (Start)
%C A370267 The set of numbers congruent to 1 modulo 8 (A017077) contains all the odd squares and generates a subgroup of the positive rational numbers (under multiplication) that contains no additional integers. The subgroup has an infinite number of cosets. The rest of the construction process extends the subgroup, reducing the number of cosets to 2, by choosing additional generators that are semiprime.
%C A370267 First we extend the subgroup to include all nonzero integer squares. As we already have the odd squares, we need only add 4, the square of the smallest prime, as a generator. The extended subgroup has only 8 cosets and its integer members are listed in A234000. To achieve a subgroup with 2 cosets we now add squarefree semiprime generators. The 2 smallest, 6 and 10, suffice.
%C A370267 The resulting subgroup has this sequence's terms as its integer members.
%C A370267 (End)
%C A370267 The equivalent process starting with numbers congruent to 1 modulo 3 (or 1 modulo 6) produces A189715. If we take its intersection with this sequence we get A370268, which starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). Similarly, if we start with numbers congruent to 1 modulo 5 (or 1 modulo 10) and take the resulting set's intersection with this sequence we get a set starting with the first 32 nonzero numbers of the form x^2 - 10y^2 (see A242664).
%C A370267 The construction process leads to a number of properties:
%C A370267 - The sequence is closed under multiplication and all integer ratios between terms are in the sequence.
%C A370267 - The sequence and its complement have the property that the terms of one can be generated by halving the even terms of the other. Each has asymptotic density 1/2.
%C A370267 Numbers whose squarefree part is congruent to {1,7} mod 8 or {6,10} mod 16.
%F A370267 {a(n) : n >= 1} = {A059897(i,j) : i in A234000, j in {1, 6, 10, 15}}.
%e A370267 7 is prime, so 7 is its only prime factor, which has the form 8m-1. So 7 has an even number (zero) of prime factors not of the form 8m+-1, and therefore is in the sequence. In terms of the subgroup generators described at the start of the comments, (13*8+1) * 4 / (6*10) = 105 * 4/60 = 7.
%e A370267 110 = 2 * 5 * 11, so it has 3 prime factors and all 3 do not have the form 8m+-1. 3 is odd, so 110 is not in the sequence.
%o A370267 (PARI) isok(k) = {c = core(k); c%8 == 1 || c%8 == 7 || c%16 == 6 || c%16 == 10}
%o A370267 (Python)
%o A370267 def A370267(n):
%o A370267     def f(x): return n+x-sum(((y:=x>>(i<<1))-7>>3)+(y-1>>3)+2 for i in range((x.bit_length()>>1)+1))-sum(((z:=x>>(i<<1)+1)-5>>3)+(z-3>>3)+2 for i in range(x.bit_length()-1>>1))
%o A370267     m, k = n, f(n)
%o A370267     while m != k: m, k = k, f(k)
%o A370267     return m # _Chai Wah Wu_, Mar 19 2025
%Y A370267 Disjoint union of A004215, A055042, A055043 and A234000.
%Y A370267 See the comments for the relationships with A002481, A017077, A189715, A242664, A370268.
%Y A370267 Cf. A042999 (primes), A059897.
%K A370267 nonn,easy
%O A370267 1,2
%A A370267 _Peter Munn_, Feb 13 2024