cp's OEIS Frontend

A364280 Lexicographically earliest sequence of distinct positive integers such that a(n) is the least novel multiple of m, the product of all primes q < gpf(a(n-2)*a(n-1)) which do not divide a(n-2)*a(n-1); a(1) = 1, a(2) = 2.

%I A364280 #16 Apr 13 2025 07:11:38
%S A364280 1,2,3,4,5,6,7,10,9,8,11,105,12,13,385,18,14,15,16,17,15015,20,19,
%T A364280 51051,30,21,22,25,42,23,230945,84,24,35,26,33,70,27,28,40,36,29,
%U A364280 37182145,48,31,1078282205,54,32,34,30030,37,6678671,60060,38,51,5005,44,39
%N A364280 Lexicographically earliest sequence of distinct positive integers such that a(n) is the least novel multiple of m, the product of all primes q < gpf(a(n-2)*a(n-1)) which do not divide a(n-2)*a(n-1); a(1) = 1, a(2) = 2.
%C A364280 It follows from the definition that the sequence is infinite.
%C A364280 Let r(n) = a(n-2)*a(n-1).
%C A364280 If rad(r(n)) is a primorial, then every prime q < gpf(r(n)) divides r(n), so m = 1, the empty product, and a(n) = u, the smallest missing number in the sequence so far.
%C A364280 If rad(r(n)) is not a primorial, then m > 1, and significant spikes can occur in scatterplot when there are multiple primes < gpf(r(n)) which do not divide r(n) (e.g., a(12) = 105, a(15) = 385, a(21) = 15015).
%C A364280 The only way a prime can occur is as u.
%C A364280 The sequence is a permutation of the positive integers since no number appears more than once and m = 1 eventually introduces any number not already placed consequent to terms arising from m > 1.
%H A364280 Michael De Vlieger, <a href="/A364280/b364280.txt">Table of n, a(n) for n = 1..6187</a>
%H A364280 Michael De Vlieger, <a href="/A364280/a364280.png">Log log scatterplot of log_10(a(n))</a>, n = 1..2^16, highlighting prime a(n) in red.
%e A364280 a(4) = 4, a(5) = 5, and 3 is the only prime < 5 which does not divide 20, therefore m = 3 and a(6) = 6 since 3 has occurred once already.
%e A364280 a(10) = 8, a(11) = 11 and the product of all primes < 11 which do not divide 8*11 = 88 is 3*5*7 = 105, which has not occurred earlier, so a(12) = 105.
%t A364280 nn = 120; c[_] := False; m[_] := 1; a[1] = i = 1; a[2] = j = 2; c[1] = c[2] = True;
%t A364280   f[x_] := Times @@ Complement[Prime@ Range[PrimePi@ #[[-1]] - 1], #] &[
%t A364280     FactorInteger[x][[All, 1]]];
%t A364280   Do[While[Set[k, f[i j]]; c[k m[k]], m[k]++]; k *= m[k];
%t A364280     Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, 3, nn}];
%t A364280 Array[a, nn] (* _Michael De Vlieger_, Jul 17 2023 *)
%Y A364280 Cf. A002110, A007947, A359804, A364154.
%K A364280 nonn
%O A364280 1,2
%A A364280 _David James Sycamore_, Jul 17 2023
%E A364280 More terms from _Michael De Vlieger_, Jul 17 2023