1,2

This is instance g = 6 of the g-family of sequences, call it r(g,n), where for g >= 2 the highest power of g is removed from n. See the crossrefs.

The present sequence is not multiplicative: a(6) = 1 not a(2)*a(3) = 6. In the prime factor decomposition one has to consider a(2^e2*3^e^3) as one entity, also for e2 >= 0, e3 >= 0 with a(1) = 1, and apply the rule given in the formula section. With this rule the sequence will be multiplicative in an unusual sense. - Wolfdieter Lang, Feb 12 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000

a(n) = n/6^A122841(n), n >= 1.

For n >= 2, a(n) is sort of multiplicative if a(2^e2*3^e3) = 2^(e2 - m)*3^(e3 - m) with m = m(e2, e3) = min(e2, e3), for e2, e3 >= 0, a(1) = 1, and a(p^e) = p^e for primes p >= 5.

From Peter Munn, Jun 04 2020: (Start)

Proximity to being multiplicative may be expressed as follows:

a(n * A007310(k)) = a(n) * a(A007310(k));

a(n^2) = a(n)^2;

a(n) = a(A007913(n)) * a(A008833(n)).

(End)

Sum_{k=1..n} a(k) ~ (3/7) * n^2. - Amiram Eldar, Nov 20 2022

a(1) = 1 = 1/6^A122841(1) = 1/6^0.

a(9) = a(2^0*3^2), min(0,2) = 0, a(9) = 2^(0-0)*3^(2-0) = 1*9 = 9.

a(12) = a(2^2*3^1), m = min(2,1) = 1, a(12) = 2^(2-1)*3^(1-1) = 2^1*1 = 2.

a(30) = a(2*3*5) = a(2^1*3^1)*a(5) = 1*a(5) = 5.

a[n_] := n/6^IntegerExponent[n, 6]; Array[a, 66] (* Robert G. Wilson v, Feb 12 2018 *)

(PARI) a(n) = n/6^valuation(n, 6); \\ Joerg Arndt, Jun 28 2014

Cf. A122841, A000265, A038502, A065883, A132739, A242603.

A007310, A007913, A008833 are used to express relationship between terms of this sequence.

Sequence in context: A325976 A375125 A319654 * A214394 A255691 A030105

Adjacent sequences: A244411 A244412 A244413 * A244415 A244416 A244417

nonn,easy

Wolfdieter Lang, Jun 27 2014

Incorrect multiplicity claim corrected by Wolfdieter Lang, Feb 12 2018

approved