A235524 Primitive refactorable (or tau) numbers: refactorable numbers which are not part of any family.

1, 2, 8, 9, 12, 18, 72, 80, 96, 108, 128, 288, 448, 625, 720, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2560, 4032, 4050, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 9408, 10800, 11250, 11264, 12960, 13122, 16200, 18000, 18432, 19440, 20412, 21952

Offset

1,2

Comments

To be "primitive", the set of prime factors of N and of d(N) must be identical, otherwise any prime only in N is arbitrary and this defines a family of refactorable numbers. These are referred to as generators in the Zelinsky reference.

This sequence is therefore the intersection of the refactorable numbers (A033950) and those numbers with identical sets of prime factors for N and d(N) (A081381).

The first numbers in A081381 which are not in this sequence are 486, 768, 8748, and 303750. This sequence is A235525.

Links

Walter Roscello and Giovanni Resta, Table of n, a(n) for n = 1..1008 (terms < 10^11, first 459 terms from Walter Roscello)

Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8

Example

720 is in the sequence since 720 = 2^4 * 3^2 * 5^1, therefore the prime decomposition of d(720) is 5 * 3 * 2 and each prime in 720 is required to make it refactorable.

Crossrefs

Cf. A000005, A033950, A081381, A235525.

Sequence in context: A336715 A120737 A081381 * A239133 A365517 A166686

Adjacent sequences: A235521 A235522 A235523 * A235525 A235526 A235527

Keyword

nonn,easy

Author

Walter Roscello, Jan 11 2014

Status

approved