cp's OEIS Frontend

A367711 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor k > 1 with A001414(a(n-1)), the sum of the primes dividing a(n-1), with repetition.

%I A367711 #21 Dec 01 2023 15:57:24
%S A367711 2,4,6,5,10,7,14,3,9,8,12,21,15,16,18,20,24,27,30,22,13,26,25,28,11,
%T A367711 33,32,34,19,38,35,36,40,44,39,42,45,55,46,50,48,66,52,17,51,54,77,56,
%U A367711 65,57,58,31,62,60,63,78,64,68,49,70,72,69,74,75,91,76,23,92,81,80,104,95,82,43,86
%N A367711 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor k > 1 with A001414(a(n-1)), the sum of the primes dividing a(n-1), with repetition.
%C A367711 In the first 100000 terms the only fixed point is 9, and it is likely no more exist. In the same range the smallest missing numbers are 4073, 5039, 5261. The sequence is conjectured to be a permutation of the integers >= 2.
%C A367711 From _Michael De Vlieger_, Nov 28 2023: (Start)
%C A367711 In scatterplot, composites fall in a cototient trajectory just above the line a(n)/n, while primes fall into several trajectories well below the line a(n)/n. This is an effect of finding the next term a(n) such that gcd(a(n), a(n-1)) = 1.
%C A367711 The trajectories T of primes a(n) arrange according to a(n+1)/a(n) = m. Hence, for example, T(m), m = 2 includes {2, 5, 7, 13, 19, 31, 43, ...}, T(3) includes {3, 11, 17} and may be finite, T(4) includes {23, 41, 47, 71, 83, 101, ...}, but T(m) for m in {5, 16, 17, ...} does not appear in the first 2^20 terms. It is evident that the trajectories T(m) are nonlinear.
%C A367711 The smallest missing number in a(1..1048576) is prime(3912) = 36899, followed by primes with indices 3995, 4151, 4179, etc. The smallest missing composite is 1116967. (End)
%H A367711 Scott R. Shannon, <a href="/A367711/b367711.txt">Table of n, a(n) for n = 1..10000</a>
%H A367711 Michael De Vlieger, <a href="/A367711/a367711_3.png">Log log scatterplot of a(n)</a>, n = 1..2^20, showing primes in red, composites in dark blue.
%H A367711 Michael De Vlieger, <a href="/A367711/a367711_2.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor prime powers in blue, accentuating numbers of the last category that are also squareful in light blue.
%H A367711 Scott R. Shannon, <a href="/A367711/a367711.png">Image of the first 100000 terms</a>. The green line is a(n) = n.
%e A367711 a(10) = 8 as a(9) = 9 and A001414(9) = 6, and 8 is the smallest unused number that shares a factor with 6. This is the first term to differ from A365060.
%t A367711 nn = 120; c[_] := False;
%t A367711   f[x_] := f[x] = Total[Times @@@ FactorInteger[x]]; f[1] = 1;
%t A367711   a[1] = j = 2; c[2] = True; u = 3;
%t A367711   Do[k = u; While[Or[c[k], CoprimeQ[j, k]], k++];
%t A367711     Set[{a[n], c[k], j}, {k, True, f[k]}];
%t A367711     If[k == u, While[c[u], u++]], {n, 2, nn}];
%t A367711 Array[a, nn] (* _Michael De Vlieger_, Nov 28 2023 *)
%Y A367711 Cf. A001414, A365059, A365060, A300813.
%K A367711 nonn
%O A367711 1,1
%A A367711 _Scott R. Shannon_, Nov 28 2023