cp's OEIS Frontend

A373357 a(1) = 1, a(2) = 2, a(3) = 15; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1), shares a factor with a(n-2), while omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

%I A373357 #15 Jun 09 2024 14:22:19
%S A373357 1,2,15,154,3,10,231,4,21,110,7,6,385,8,33,70,9,14,165,16,35,66,5,12,
%T A373357 455,27,20,273,25,18,595,32,45,182,81,22,105,11,24,715,64,39,140,13,
%U A373357 28,195,49,26,315,128,51,130,17,36,935,243,34,285,256,55,42,121,38,429,19,44,399,512,57,170
%N A373357 a(1) = 1, a(2) = 2, a(3) = 15; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1), shares a factor with a(n-2), while omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).
%C A373357 The sequence uses the same rules for selecting the next term as the Yellowstone permutation A098550 but with the additional restriction that the number of distinct prime factors of a(n) must be different to both a(n-1) and a(n-2). The terms show complicated behavior, being concentrated along various curved and straight lines some of which cross and some of which only have points for various ranges of n. See the attached images.
%C A373357 The fixed points begin 1, 2, 32, 51, although it is possible more exist. The sequence is likely to be a permutation of the positive integers.
%H A373357 Scott R. Shannon, <a href="/A373357/b373357.txt">Table of n, a(n) for n = 1..10000</a>
%H A373357 Michael De Vlieger, <a href="/A373357/a373357_2.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, where purple additionally represents powerful numbers that are not prime powers.
%H A373357 Scott R. Shannon, <a href="/A373357/a373357.png">Image of the first 5000 points</a>. Numbers with one, two, three, or four and more distinct prime factors are show as red, yellow, green and violet respectively. The white line is a(n) = n.
%H A373357 Scott R. Shannon, <a href="/A373357/a373357_1.png">Image of the first 200000 points</a>.
%e A373357 a(10) = 110 as 110 shares a factor with a(8) = 4, does not share a factor with a(9) = 21, while omega(110) = 3 does not equal omega(4) = 1 or omega(21) = 2.
%t A373357 nn = 63; c[_] := False;
%t A373357 MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 2, 15}];
%t A373357 i = a[2]; j = a[3]; u = 3; v = 1; w = 2;
%t A373357 Do[k = u;
%t A373357  While[Or[c[k],
%t A373357    ! CoprimeQ[j, k],
%t A373357    ! DuplicateFreeQ[{v, w, Set[x, PrimeNu[k]]}]],
%t A373357  k++];
%t A373357  Set[{a[n], c[k], i, j, v, w}, {k, True, j, k, w, x}];
%t A373357  If[k == u, While[c[u], u++]], {n, 4, nn}];
%t A373357 Array[a, nn] (* _Michael De Vlieger_, Jun 09 2024 *)
%Y A373357 Cf. A098550, A001221, A373350, A372975, A372974.
%K A373357 nonn
%O A373357 1,2
%A A373357 _Scott R. Shannon_, Jun 02 2024