%I #20 Jul 20 2024 20:09:26
%S 9,18,25,27,36,45,49,50,54,55,63,72,75,81,90,98,99,100,108,110,117,
%T 121,125,126,135,144,147,150,162,165,169,175,180,196,198,200,216,220,
%U 225,234,242,243,245,250,252,270,275,288,289,294,300,315,324,325,330,338,343,350,360,361,363,375,385,392,396,400
%N Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).
%C Provided there do not exist any odd perfect numbers, these are numbers k for which A347240(k) >= A006530(k), as for any odd perfect number x, A347240(x) = -1 by its escape clause.
%C If k is included as a term, then 2*k is also present.
%C Not all odd squares of primes are present. For example, 67^2 and 79^2 are not included. See also A091490, which seems to be a subsequence of those exceptions.
%C Conjecture: There are no primes in this sequence. Checked up to the 2^20-th prime, 16290047.
%H <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%e For n = 55 = 5*11, on the first iteration we get A000593(55) = 72 = 2^3 * 3^2, but both 2 and 3 are less than 11; therefore we iterate a second time to get A000593(72) = 13, which is the first value whose largest prime factor is larger than that of 55 (13 > 11), thus 55 is included in the sequence.
%o (PARI)
%o A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
%o A000265(n) = (n >> valuation(n, 2));
%o A000593(n) = sigma(A000265(n));
%o A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf,return(1))); (0); };
%o isA347242(n) = A347244(n);
%Y Cf. A000593, A006530, A091490, A161942, A336353, A336361, A347240, A347241, A347243 (complement), A347244 (characteristic function).
%Y Positions of nonzero terms in A347245.
%K nonn
%O 1,1
%A _Antti Karttunen_, Aug 28 2021