|
%I #22 Nov 18 2022 03:38:46
%S 1,2,4,6,3,9,12,8,10,5,15,18,14,7,21,24,16,20,25,30,36,42,28,56,70,35,
%T 105,27,33,11,22,26,13,39,45,40,32,34,17,51,48,38,19,57,54,44,55,50,
%U 46,23,69,60,80,90,63,49,77,66,72,78,52,104,130,65,195,75,120
%N a(1)=1, a(2)=2. Thereafter, if there are prime divisors p | a(n-1) that are coprime to a(n-2), a(n) is the least novel multiple of the product of these primes. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-1). See comments.
%C Let k be the greatest common squarefree divisor of a(n-2) and a(n-1) and let s = A007947(a(n-1)). If k = 1, then a(n) = m_s*s, else a(n) = m_k*k, where m_i is the smallest multiple of i such that m*i does not appear in a(1..n-1).
%C Variant of A357963; a(21) = 36, but A357963(21) = 22.
%H Michael De Vlieger, <a href="/A357942/b357942.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A357942/a357942.png">Scatterplot of a(n)</a>, n = 1..2^20.
%H Michael De Vlieger, <a href="/A357942/a357942_1.png">Log-log scatterplot of a(n)</a>, n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green and other prime powers in gold.
%e a(1) = 1, a(2) = 2 and 2 divides 2 but does not divide 1. Since 2 is the only prime divisor of 2, a(3) = 4, the smallest unused multiple of 2.
%e Since every prime divisor of a(3)=4 also divides a(2) = 2, a(4) = 6, the least novel multiple of the squarefree kernel of 4.
%e a(19,20) = (25,30); 2|30 and 3|30 but 2 and 3 do not divide 25. The smallest multiple of 2*3 = 6 not already in the sequence is 36. Therefore a(21) = 36.
%t nn = 67; c[_] = False; q[_] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; q[2] = 2; Do[m = FactorInteger[a[n - 1]][[All, 1]]; If[Length@ # == 0, While[Set[k, #* q[#]]; c[k], q[#]++] &[Times @@ m], While[Set[k, #* q[#]]; c[k], q[#]++] &[Times @@ #]] &@ Select[m, CoprimeQ[#, a[n - 2]] &]; Set[{a[n], c[k]}, {k, True}], {n, 3, nn}]; Array[a, nn]
%Y Cf. A007947, A064413, A357963.
%K nonn
%O 1,2
%A _Michael De Vlieger_, Oct 22 2022
|
