login
A347242
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).
7
9, 18, 25, 27, 36, 45, 49, 50, 54, 55, 63, 72, 75, 81, 90, 98, 99, 100, 108, 110, 117, 121, 125, 126, 135, 144, 147, 150, 162, 165, 169, 175, 180, 196, 198, 200, 216, 220, 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
OFFSET
1,1
COMMENTS
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.
If k is included as a term, then 2*k is also present.
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.
Conjecture: There are no primes in this sequence. Checked up to the 2^20-th prime, 16290047.
EXAMPLE
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.
PROG
(PARI)
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf, return(1))); (0); };
isA347242(n) = A347244(n);
CROSSREFS
Cf. A000593, A006530, A091490, A161942, A336353, A336361, A347240, A347241, A347243 (complement), A347244 (characteristic function).
Positions of nonzero terms in A347245.
Sequence in context: A015798 A028494 A167663 * A038838 A347247 A038837
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 28 2021
STATUS
approved