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%I A349188 #39 Mar 31 2023 09:18:11
%S A349188 2,24,48,240,1440,2400,7440,25920,72000,234000
%N A349188 Largest n-phobe number.
%C A349188 A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n such that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n. A number that is not n-phile is called n-phobe.
%C A349188 The idea for this sequence and the words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).
%C A349188 The number of n-phobe numbers is always finite (A349189), the smallest one is always 1, and this sequence lists the largest n-phobe numbers.
%C A349188 a(6) >= 720. - _Michel Marcus_, Nov 14 2021
%C A349188 From _David A. Corneth_, Nov 14 2021
%C A349188 a(6) >= 1440. If a(6) > 1440 then a(6) > 50000.
%C A349188 a(7) >= 2400. If a(7) > 2400 then a(7) > 50000.
%C A349188 a(8) >= 7440. If a(8) > 7440 then a(8) > 100000.
%C A349188 a(9) >= 25920. If a(9) > 25920 then a(9) > 100000. (End)
%C A349188 Indeed, all these bounds are the corresponding values of a(6), a(7), a(8), a(9). Proof in link. For n >= 5, the five known terms are divisible by 240. - _Bernard Schott_, Nov 19 2021
%H A349188 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a4-equations-diophantiennes/3143-a496-pentaphiles-et-pentaphobes">A496 - Pentaphiles et pentaphobes</a> (in French).
%H A349188 Bernard Schott, <a href="/A349188/a349188_1.txt">Proof that a(6) = 1440 and other proofs</a>.
%e A349188 For n = 2, integers 1 and 2 are 2-phobe, then for m >= 3, every m = 1 + (m-1) with 1 < m-1 and 1 divides m-1, so, each m >= 3 is 2-phile number and a(2) = 2.
%Y A349188 Cf. A349189.
%Y A349188 k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
%Y A349188 k-phobe numbers: A019532 (k=3), A348519 (k=4), A348520 (k=5).
%K A349188 nonn,more
%O A349188 2,1
%A A349188 _Bernard Schott_, Nov 09 2021
%E A349188 a(6)..a(11) from _David A. Corneth_, Nov 19 2021

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