A383174 Permutation of the natural numbers formed by ordering by max(gpfi,bigomega), then bigomega, then numerically, where gpfi(k) = A061395(k) and bigomega(k) = A001222(k).

1, 2, 3, 4, 6, 9, 5, 10, 15, 25, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 7, 14, 21, 35, 49, 28, 42, 63, 70, 98, 105, 147, 175, 245, 343, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525

Offset: 1

Author

Bassam Abdul-Baki, Apr 18 2025

Keywords

• nonn

Comments

The sequence can be constructed starting with term 1 and then:

At step number m >= 1, append terms k with gpfi(k) <= m and bigomega(k) <= m, and which have not already appeared, and ordered first by bigomega and then numerically.

Terms which have not appeared are exactly those with gpfi(k) = m or bigomega(k) = m, and in particular they start with prime(m) and end with prime(m)^m.

First differs from A344844 at n=30, with its ordering by prime exponents differing from here ordering numerically. - Michael S. Branicky, Apr 20 2025

Examples

			At step m=2, the new terms added are 3, 4, 6, 9, being those with gpfi(k) = 2, or with bigomega(k) = 2.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..12870 (all terms involving only primes <= 19, so ending with 19^8)
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A061395, A001222, A263297, A344844.

Programs

Python

from math import prod
from sympy import nextprime
from itertools import count, islice, combinations_with_replacement as cwr
def agen(): # generator of terms
    aset, plst = set(), [1, 2]
    for n in count(1):
        row = []
        for mc in cwr(plst, n):
            p = prod(mc)
            if p not in aset:
                row.append((n-mc.count(1), p))
                aset.add(p)
        plst.append(nextprime(plst[-1]))
        yield from (p for m, p in sorted(row))
print(list(islice(agen(), 80))) # Michael S. Branicky, Apr 18 2025

Extensions

a(33) and on corrected by Michael S. Branicky, Apr 19 2025