A113552 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.

1, 2, 6, 3, 12, 4, 20, 5, 10, 30, 15, 60, 420, 7, 14, 42, 21, 84, 28, 140, 35, 70, 210, 105, 630, 9, 18, 90, 45, 180, 36, 252, 63, 126, 1260, 315, 1890, 27, 54, 270, 135, 540, 108, 756, 189, 378, 3780, 945, 5670, 81, 162, 810, 405, 1620, 324, 2268, 567, 1134, 11340, 2835

Offset: 1

Author

Amarnath Murthy, Nov 03 2005

Keywords

• easy
• nonn

Comments

From Michael De Vlieger, May 18 2018: (Start)

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.

n (mod 12) k or d 2 3 5 7

-------------------------------------------

0 ODD 3^(e-1) 5 7

1 6 2 3^e 5 7

2 ODD 3^e

3 2 2 3^e

4 5 2 3^e 5

5 ODD 3^e 5

6 4 2^2 3^e 5

7 EVEN 2^2 3^e

8 7 2^2 3^e 7

9 ODD 3^e 7

10 2 2 3^e 7

11 10 2^2 3^e 5 7

Conjectures:

1. All terms are divisible only by some combination of the smallest 4 primes; A113553 is finite at four terms.

2. For n > 24 such that n (mod 12) = 2, a(n) = 3^((n - 2)/12). (End)

Examples

			After 4 the next term is 20 and not 8 as 8 and 4 have the same prime divisors.
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
		

Links

• Antti Karttunen & Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Notes on A113552

Crossrefs

Differs from A282291 for the first time at n=25, where a(25) = 630, while A282291(25) = 840.

Differs from A304752 for the first time at n=28, where a(28) = 90, while A113552(28) = 72.

Programs

Maple

S:= {1}: A[1]:= 1:
for n from 2 to 60 do
  d:= min(numtheory:-divisors(A[n-1]) minus S);
  if d < infinity then A[n]:= d
  else
    Q:= numtheory:-factorset(A[n-1]);
    for k from 2 do
      if not member(k*A[n-1], S) and not (numtheory:-factorset(k) subset Q) then
        A[n]:= k*A[n-1];
        break
      fi
    od
  fi;
  S:= S union {A[n]}
od: # Robert Israel, May 22 2018

Mathematica

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 *)

PARI

up_to = (2^14)+1;
A007947(n) = factorback(factorint(n)[, 1]);
v113552 = vector(up_to);
m_occurrences = Map();
k=0; prev=1;
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]);
A113552(n) = v113552[n]; \\ Antti Karttunen, May 18 2018

Extensions

More terms from Robert G. Wilson v, Aug 20 2006

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