%I #53 Feb 12 2022 13:32:17
%S 1,2,6,3,12,4,20,5,10,30,15,60,420,7,14,42,21,84,28,140,35,70,210,105,
%T 630,9,18,90,45,180,36,252,63,126,1260,315,1890,27,54,270,135,540,108,
%U 756,189,378,3780,945,5670,81,162,810,405,1620,324,2268,567,1134,11340,2835
%N Beginning with 1, least divisor of the previous term not included earlier, otherwise the least multiple of the previous term having at least one prime divisor coprime to it and not included earlier.
%C From _Michael De Vlieger_, May 18 2018: (Start)
%C In the table below, we note a cycle that exists for n >= 24 and at least through n = 2^14. Let e = floor(n/12). We write multiple k if the condition is false, or the parity of divisor d if d does not occur in a. We can express a(n) as the product of the smallest four primes as shown below.
%C n (mod 12) k or d 2 3 5 7
%C -------------------------------------------
%C 0 ODD 3^(e-1) 5 7
%C 1 6 2 3^e 5 7
%C 2 ODD 3^e
%C 3 2 2 3^e
%C 4 5 2 3^e 5
%C 5 ODD 3^e 5
%C 6 4 2^2 3^e 5
%C 7 EVEN 2^2 3^e
%C 8 7 2^2 3^e 7
%C 9 ODD 3^e 7
%C 10 2 2 3^e 7
%C 11 10 2^2 3^e 5 7
%C Conjectures:
%C 1. All terms are divisible only by some combination of the smallest 4 primes; A113553 is finite at four terms.
%C 2. For n > 24 such that n (mod 12) = 2, a(n) = 3^((n - 2)/12). (End)
%H Antti Karttunen & Michael De Vlieger, <a href="/A113552/b113552.txt">Table of n, a(n) for n = 1..16384</a>
%H Michael De Vlieger, <a href="/A113552/a113552.txt">Notes on A113552</a>
%e After 4 the next term is 20 and not 8 as 8 and 4 have the same prime divisors.
%e After a(27) = 18 = 2 * 3^2, the next term a(28) is neither 2*18 = 2^2 * 3^2, nor 3*18 = 2 * 3^3 nor 4*18 = 2^3 * 3^2 as none of them have any prime factors that would not occur already in 18. But 5*18 = 90 has such a factor, and 90 has not occurred before, thus a(28) = 90. - _Antti Karttunen_, May 18 2018
%p S:= {1}: A[1]:= 1:
%p for n from 2 to 60 do
%p d:= min(numtheory:-divisors(A[n-1]) minus S);
%p if d < infinity then A[n]:= d
%p else
%p Q:= numtheory:-factorset(A[n-1]);
%p for k from 2 do
%p if not member(k*A[n-1], S) and not (numtheory:-factorset(k) subset Q) then
%p A[n]:= k*A[n-1];
%p break
%p fi
%p od
%p fi;
%p S:= S union {A[n]}
%p od: # _Robert Israel_, May 22 2018
%t Nest[Append[#, Block[{d = Complement[Divisors@ #[[-1]], #], k = 2}, If[d != {}, d[[1]], While[Nand[PowerMod[#[[-1]], k, k] != 0, FreeQ[#, k #[[-1]] ] ], k++]; k #[[-1]] ] ] ] &, {1}, 116] (* _Robert G. Wilson v_, Aug 20 2006, corrected by _Michael De Vlieger_, May 18 2018 *)
%o (PARI)
%o up_to = (2^14)+1;
%o A007947(n) = factorback(factorint(n)[, 1]);
%o v113552 = vector(up_to);
%o m_occurrences = Map();
%o k=0; prev=1;
%o for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m_occurrences,d),v113552[n] = d;mapput(m_occurrences,d,n);break)); if(!v113552[n], m = 1; try = prev; while(!(prev%A007947(m)) || mapisdefined(m_occurrences,try), m++; try = prev*m); mapput(m_occurrences,v113552[n] = try,n)); prev = v113552[n]);
%o A113552(n) = v113552[n]; \\ _Antti Karttunen_, May 18 2018
%Y Differs from A282291 for the first time at n=25, where a(25) = 630, while A282291(25) = 840.
%Y Differs from A304752 for the first time at n=28, where a(28) = 90, while A113552(28) = 72.
%K easy,nonn
%O 1,2
%A _Amarnath Murthy_, Nov 03 2005
%E More terms from _Robert G. Wilson v_, Aug 20 2006
%E Sequence's terms changed back to the original author's intended meaning, differing from those computed by Wilson and Johnston from n = 28 onward, with a(28) = 90 instead of a(28) = 72. The latter version was recreated with the new A-number A304752, to which also the old Maple and Mathematica programs were transferred. - _Antti Karttunen_, May 18 2018