A244414 Remove highest power of 6 from n.

1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 2, 13, 14, 15, 16, 17, 3, 19, 20, 21, 22, 23, 4, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 7, 43, 44, 45, 46, 47, 8, 49, 50, 51, 52, 53, 9, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 11

Offset

1,2

Comments

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A325959 A325976 A319654 * A214394 A255691 A030105

Adjacent sequences: A244411 A244412 A244413 * A244415 A244416 A244417

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 27 2014

Extensions

Incorrect multiplicity claim corrected by Wolfdieter Lang, Feb 12 2018

Status

approved