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

Author

Wolfdieter Lang, Jun 27 2014

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

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

Crossrefs

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

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

Programs

Mathematica

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

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

Extensions

Incorrect multiplicity claim corrected by Wolfdieter Lang, Feb 12 2018