A303762 a(0) = 1, and for n >= 1, a(n) is either the largest divisor of a(n-1) not already present in the sequence, or (if all divisors already used), a(n-1) * {the least prime p such that p does not divide a(n-1) and p*a(n-1) is not already present}.

1, 2, 6, 3, 15, 5, 10, 30, 210, 105, 35, 7, 14, 42, 21, 231, 77, 11, 22, 66, 33, 165, 55, 110, 330, 2310, 1155, 385, 770, 154, 462, 6006, 3003, 1001, 143, 13, 26, 78, 39, 195, 65, 130, 390, 2730, 1365, 455, 91, 182, 546, 273, 4641, 1547, 221, 17, 34, 102, 51, 255, 85, 170, 510, 3570, 1785, 595, 119, 238, 714, 357, 3927, 1309, 187, 374, 1122, 561, 2805, 935

Offset

0,2

Comments

Each a(n+1) is either a divisor or a multiple of a(n).

The construction is otherwise like that of A303760, except here we choose the largest divisor instead of the smallest one. In contrast to A303760, this sequence is NOT permutation of A005117: 70 = A019565(13) is the first missing squarefree number. See also comments in A303769, A303749 and A302775.

Index of greatest prime factor of a(n) is monotonic and increments at n = {0, 1, 2, 4, 8, 15, 31, 50, 102, 157, 317, 480, 964, 1451, 2907, 4366, 8738, 13113, 26233, 39356, ...} - Michael De Vlieger, May 22 2018

Links

Antti Karttunen, Table of n, a(n) for n = 0..26232

Formula

a(n) = A019565(A303769(n)). [Conjectured]

Example

From Michael De Vlieger, May 23 2018: (Start)
Table below shows the initial 31 terms at right. First column is index n. Second shows "." if a(n) = largest divisor of a(n-1), or factor p. Third shows presence "1" or absence "." of prime k among prime divisors of a(n).
   n   p\d    MN(n)      a(n)
  ---------------------------
   0     .    .            1
   1     2    1            2
   2     3    11           6
   3     .    .1           3
   4     5    .11         15
   5     .    ..1          5
   6     2    1.1         10
   7     3    111         30
   8     7    1111       210
   9     .    .111       105
  10     .    ..11        35
  11     .    ...1         7
  12     2    1..1        14
  13     3    11.1        42
  14     .    .1.1        21
  15    11    .1.11      231
  16     .    ...11       77
  17     .    ....1       11
  18     2    1...1       22
  19     3    11..1       66
  20     .    .1..1       33
  21     5    .11.1      165
  22     .    ..1.1       55
  23     2    1.1.1      110
  24     3    111.1      330
  25     7    11111     2310
  26     .    .1111     1155
  27     .    ..111      385
  28     2    1.111      770
  29     .    1..11      154
  30     3    11.11      462
  31    13    11.111    6006
  ...  (End)

Mathematica

Nest[Append[#, Block[{d = Divisors@ #[[-1]], p = 2}, If[Complement[d, #] != {}, Complement[d, #][[-1]], While[Nand[Mod[#[[-1]], p] != 0, FreeQ[#, p #[[-1]] ] ], p = NextPrime@ p]; p #[[-1]] ] ] ] &, {1}, 75] (* Michael De Vlieger, May 22 2018 *)

Prog

(PARI)
default(parisizemax, 2^31);
up_to = 2^14;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303762 = vector(up_to);
m_inverses = Map();
prev=1; for(n=1, up_to, fordiv(prev, d, if(!mapisdefined(m_inverses, (prev/d)), v303762[n] = (prev/d); mapput(m_inverses, (prev/d), n); break)); if(!v303762[n], apu = prev; while(mapisdefined(m_inverses, try = prev*A053669(apu)), apu *= A053669(apu)); v303762[n] = try; mapput(m_inverses, try, n)); prev = v303762[n]);
A303762(n) = v303762[n+1];

Crossrefs

Subset of A005117.

Cf. also A019565, A302774, A302775, A303749, A303769.

Cf. A303760, A303761 (variants).

Sequence in context: A303760 A330919 A303770 * A303778 A094426 A094302

Adjacent sequences: A303759 A303760 A303761 * A303763 A303764 A303765

Keyword

nonn

Author

Antti Karttunen, May 03 2018

Status

approved