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

1,2

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

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

Example

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

Prog

(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

Crossrefs

Cf. A061395, A001222, A263297, A344844.

Sequence in context: A378194 A233559 A285332 * A344844 A185290 A363995

Adjacent sequences: A383171 A383172 A383173 * A383175 A383176 A383177

Keyword

nonn

Author

Bassam Abdul-Baki, Apr 18 2025

Extensions

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

Status

approved