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

Author

Walter Roscello, Jan 11 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

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

Crossrefs

Cf. A000005, A033950, A081381, A235525.