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

Author

Antti Karttunen, May 02 2018

Keywords

• nonn

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

Crossrefs

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

Cf. also A303751, A303760, A303762.

Programs

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];

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).