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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}.
7
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
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. A303760, A303761 (variants).
Sequence in context: A303760 A330919 A303770 * A303778 A094426 A094302
KEYWORD
nonn
AUTHOR
Antti Karttunen, May 03 2018
STATUS
approved