A303761 Divisor-or-multiple permutation of squarefree numbers: a(0) = 1, and for n >= 1, a(n) is either the least divisor of a(n-1) not already present, or (if all divisors already used), a(n) is obtained by iterating the map x -> x*A053669(x), starting from x = a(n-1), until x is found which is not already present in the sequence.

1, 2, 6, 3, 30, 5, 10, 210, 7, 14, 42, 21, 2310, 11, 22, 66, 33, 330, 15, 30030, 13, 26, 78, 39, 390, 65, 130, 2730, 35, 70, 510510, 17, 34, 102, 51, 510, 85, 170, 3570, 105, 9699690, 19, 38, 114, 57, 570, 95, 190, 3990, 133, 266, 798, 399, 43890, 55, 110, 223092870, 23, 46, 138, 69, 690, 115, 230, 4830, 161, 322, 966, 483, 53130, 77

Offset

0,2

Comments

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

The primorials (A002110) occur in ascending order, in positions given by A300829, and each is then followed by the least unused term up to that point. For n = 2 .. 79 this is the highest prime factor of the said primorial, but note that for A300829(80) = 4965, a(4965) = A002110(80), but a(4966) = 407 = 11*37, instead of prime(80) = 409. Note that 409 occurs at a(5043), where 5043 = 1+A300829(81).

For example, 11 comes after a(A300829(5)) = a(12) = 2310 = 2*3*5*7*11, and all squarefree numbers < 11: {1, 2, 3, 5, 6, 7, 10} occur before a(13).

Links

Antti Karttunen, Table of n, a(n) for n = 0..4095

Formula

a(n) = A019565(A303765(n)).

For n >= 0, a(A300829(n)) = A002110(n) [primorials are the records].

For n = 2 .. 79, a(1+A300829(n)) = A000040(n).

Prog

(PARI)
default(parisizemax, 2^31);
up_to = 2^8;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303761 = vector(up_to);
m_inverses = Map();
prev=1; for(n=1, up_to, fordiv(prev, d, if(!mapisdefined(m_inverses, d), v303761[n] = d; mapput(m_inverses, d, n); break)); if(!v303761[n], while(mapisdefined(m_inverses, prev), prev *= A053669(prev)); v303761[n] = prev; mapput(m_inverses, prev, n)); prev = v303761[n]);
A303761(n) = v303761[n+1];

Crossrefs

Cf. A005117, A019565, A053669, A300829 (gives the positions of records), A303765.

Cf. also A303751, A303760, A303762.

Sequence in context: A331431 A248120 A144362 * A283478 A125666 A371799

Adjacent sequences: A303758 A303759 A303760 * A303762 A303763 A303764

Keyword

nonn

Author

Antti Karttunen, May 02 2018

Status

approved