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A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details). 4
1, 2, 4, 6, 8, 12, 16, 18, 30, 36, 40, 48, 60, 64, 72, 81, 90, 108, 144, 162, 180, 192, 200, 210, 225, 240, 256, 280, 320, 324, 360, 405, 420, 432, 450, 500, 512, 540, 576, 600, 630, 648, 720, 768, 810, 900, 960, 1260, 1280, 1296, 1350, 1400, 1536, 1575, 1600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The prime tower factorization of a number is defined in A182318.

We say that two numbers, say n and m, belong to the same prime tower factorization equivalence class iff there is a permutation of the prime numbers, say f, such that replacing each prime p by f(p) in the prime tower factorization of n leads to m.

The notion of prime tower factorization equivalence class can be seen as a generalization of the notion of prime signature; thereby, this sequence can be seen as an equivalent of A025487.

This sequence contains all primorial numbers (A002110).

This sequence contains A260548.

This sequence contains the terms > 0 in A014221.

If n appears in the sequence, then 2^n appears in the sequence.

If n appears in the sequence and k>=0, then A002110(k)^n appears in the sequence.

With the exception of term 1, this sequence contains no term from A182318.

Odd numbers appearing in this sequence: 1, 81, 225, 405, 1575, 2025, 2835, 6125, 10125, 11025, 14175, 15625, 16875, 17325, 31185, 33075, 50625, 67375, 70875, 99225, ...

Here are some prime tower factorization equivalence classes:

- Class 1: the number one (the only finite equivalence class),

- Class p: the prime numbers (A000040),

- Class p*q: the squarefree semiprimes (A006881),

- Class p^p: the numbers of the form p^p with p prime (A051674),

- Class p^q: the numbers of the form p^q with p and q distinct primes,

- Class p*q*r: the sphenic numbers (A007304),

- Class p*q*r*s: the products of four distinct primes (A046386),

- Class p*q*r*s*t: the products of five distinct primes (A046387),

- Class p*q*r*s*t*u: the products of six distinct primes (A067885).

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..1000

Rémy Sigrist, PARI program for A279686

Rémy Sigrist, Prime tower factorization of the first terms

EXAMPLE

2 is the least number of the form p with p prime, hence 2 appears in the sequence.

6 is the least number of the form p*q with p and q distinct primes, hence 6 appears in the sequence.

72 is the least number of the form p^q*q^p with p and q distinct primes, hence 72 appears in the sequence.

36000 is the least number of the form p^q*q^r*r^p with p, q and r distinct primes, hence 36000 appears in the sequence.

CROSSREFS

Cf. A000040, A002110, A006881, A007304, A014221, A025487, A046386, A046387, A051674, A067885, A182318, A260548, A279690.

Sequence in context: A322492 A332276 A317246 * A219653 A050622 A082662

Adjacent sequences:  A279683 A279684 A279685 * A279687 A279688 A279689

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Dec 16 2016

STATUS

approved

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)