cp's OEIS Frontend

A320123 a(1) = 1, a(2) = 2, a(3) = 3, and for any n > 3, a(n) = the smallest positive integer not yet in the sequence such that gcd(a(n-2), a(n-1)), gcd(a(n-1), a(n)) and gcd(a(n), a(n-2)) are all distinct.

%I A320123 #11 Oct 13 2018 02:37:17
%S A320123 1,2,3,6,4,9,12,8,15,10,14,5,20,16,25,30,18,27,22,24,11,33,21,7,36,28,
%T A320123 35,26,40,13,52,32,39,42,34,17,38,68,19,76,44,55,45,48,46,23,50,92,60,
%U A320123 51,56,54,49,63,57,70,66,65,75,69,80,72,78,64,81,84,58
%N A320123 a(1) = 1, a(2) = 2, a(3) = 3, and for any n > 3, a(n) = the smallest positive integer not yet in the sequence such that gcd(a(n-2), a(n-1)), gcd(a(n-1), a(n)) and gcd(a(n), a(n-2)) are all distinct.
%C A320123 This sequence is a variant of A127202.
%C A320123 The sequence is well defined as for any n > 3, provided the first n terms are known, any number v of the form a(n-2) * b where b is coprime to a(n) * a(n-1) satisfies #{ gcd(a(n-1), a(n)), gcd(a(n), v), gcd(v, a(n-1)) } = #{ gcd(a(n-1), a(n)), gcd(a(n), a(n-2)), gcd(a(n-2), a(n-1)) } = 3, and a(n+1) exists.
%C A320123 In the scatterplot of the sequence, the prime numbers correspond to the lower line.
%H A320123 Rémy Sigrist, <a href="/A320123/b320123.txt">Table of n, a(n) for n = 1..10000</a>
%H A320123 Rémy Sigrist, <a href="/A320123/a320123.gp.txt">PARI program for A320123</a>
%e A320123 The first terms, alongside gcd(a(n-2), a(n-1)), gcd(a(n-1), a(n)) and gcd(a(n), a(n-2)), are:
%e A320123   n   a(n)  gcd(a(n-2),a(n-1))  gcd(a(n-1),a(n))  gcd(a(n),a(n-2))
%e A320123   --  ----  ------------------  ----------------  ----------------
%e A320123    1     1                 N/A               N/A               N/A
%e A320123    2     2                 N/A                 1               N/A
%e A320123    3     3                   1                 1                 1
%e A320123    4     6                   1                 3                 2
%e A320123    5     4                   3                 2                 1
%e A320123    6     9                   2                 1                 3
%e A320123    7    12                   1                 3                 4
%e A320123    8     8                   3                 4                 1
%e A320123    9    15                   4                 1                 3
%e A320123   10    10                   1                 5                 2
%e A320123   11    14                   5                 2                 1
%e A320123   12     5                   2                 1                 5
%e A320123   13    20                   1                 5                 2
%e A320123   14    16                   5                 4                 1
%e A320123   15    25                   4                 1                 5
%o A320123 (PARI) See Links section.
%Y A320123 Cf. A127202.
%K A320123 nonn
%O A320123 1,2
%A A320123 _Rémy Sigrist_, Oct 06 2018