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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).

%I A383174 #55 May 02 2025 22:28:45
%S A383174 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,
%T A383174 28,42,63,70,98,105,147,175,245,343,16,24,36,40,54,56,60,81,84,90,100,
%U A383174 126,135,140,150,189,196,210,225,250,294,315,350,375,441,490,525
%N 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).
%C A383174 The sequence can be constructed starting with term 1 and then:
%C A383174 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.
%C A383174 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.
%C A383174 First differs from A344844 at n=30, with its ordering by prime exponents differing from here ordering numerically. - _Michael S. Branicky_, Apr 20 2025
%H A383174 Michael S. Branicky, <a href="/A383174/b383174.txt">Table of n, a(n) for n = 1..12870</a> (all terms involving only primes <= 19, so ending with 19^8)
%H A383174 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A383174 At step m=2, the new terms added are 3, 4, 6, 9, being those with gpfi(k) = 2, or with bigomega(k) = 2.
%o A383174 (Python)
%o A383174 from math import prod
%o A383174 from sympy import nextprime
%o A383174 from itertools import count, islice, combinations_with_replacement as cwr
%o A383174 def agen(): # generator of terms
%o A383174     aset, plst = set(), [1, 2]
%o A383174     for n in count(1):
%o A383174         row = []
%o A383174         for mc in cwr(plst, n):
%o A383174             p = prod(mc)
%o A383174             if p not in aset:
%o A383174                 row.append((n-mc.count(1), p))
%o A383174                 aset.add(p)
%o A383174         plst.append(nextprime(plst[-1]))
%o A383174         yield from (p for m, p in sorted(row))
%o A383174 print(list(islice(agen(), 80))) # _Michael S. Branicky_, Apr 18 2025
%Y A383174 Cf. A061395, A001222, A263297, A344844.
%K A383174 nonn
%O A383174 1,2
%A A383174 _Bassam Abdul-Baki_, Apr 18 2025
%E A383174 a(33) and on corrected by _Michael S. Branicky_, Apr 19 2025