A260548 -id:A260548 - cp's OEIS frontend

A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details).

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

Offset: 1

Rémy Sigrist, Dec 16 2016

nonn

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).

			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.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Roberto Conti and Pierluigi Contucci, A Natural Avenue, arXiv:2204.08982 [math.NT], 2022.
Rémy Sigrist, PARI program for A279686
Rémy Sigrist, Prime tower factorization of the first terms

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

A284889 Numbers n such that A279513(n) is a primorial number (A002110).

Original entry on oeis.org

1, 2, 6, 8, 9, 30, 40, 45, 75, 96, 210, 250, 280, 315, 486, 525, 672, 735, 1750, 1920, 2310, 3080, 3402, 3430, 3465, 5775, 6125, 7392, 8085, 8575, 10976, 11907, 12705, 15625, 16000, 19250, 21120, 21870, 30030, 31104, 32768, 37422, 37730, 40040, 45045, 54675

Offset: 1

Rémy Sigrist, Apr 05 2017

nonn

Also numbers with the k first prime numbers in their prime tower factorization, without duplicate, for some k (see A182318 for the definition of the prime tower factorization of a number).
This sequence contains the primorial numbers (A002110); 8 = 2^3 is the first term in this sequence that is not a primorial number.
This sequence contains A260548.
All terms belong to A284763.
If a(n) <= p# for some prime p, then a(n) is p-smooth (p# denotes the product of the primes <= p, see A002110).
There are A000272(k+1) terms with k prime numbers in their prime tower factorization:
- for k=0: 1,
- for k=1: 2,
- for k=2: 2*3, 2^3, 3^2,
- for k=3: 2*3*5, 2^3*5, 2^5*3, 3^2*5, 3^5*2, 5^2*3, 5^3*2, 2^(3*5), 3^(2*5), 5^(2*3), 2^3^5, 2^5^3, 3^2^5, 3^5^2, 5^2^3, 5^3^2.

			1626625 = 5^3*7*11*13^2 appears in this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..500
Rémy Sigrist, Illustration of the first terms

Cf. A000272, A002110, A182318, A260548, A279513, A284763.

isprimorial(n) = if (n==1, 1, my (f=factor(n)); (#f~ == primepi(vecmax(f[,1]))) && (vecmax(f[,2]) == 1));
a279513(n) =  my (f=factor(n)); prod(i=1, #f~, f[i, 1]*a279513(f[i, 2]));
isok(n) = isprimorial(a279513(n)); \\ Michel Marcus, Apr 08 2017

cp's OEIS Frontend

A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs