cp's OEIS Frontend

A348517 Positive integers m with the property that there are 4 positive integers b_1 < b_2 < b_3 < b_4 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, and m = b_1 + b_2 + b_3 + b_4.

%I A348517 #16 Oct 22 2021 23:52:22
%S A348517 15,19,21,22,23,27,28,29,30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,
%T A348517 46,47,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,
%U A348517 70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92
%N A348517 Positive integers m with the property that there are 4 positive integers b_1 < b_2 < b_3 < b_4 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, and m = b_1 + b_2 + b_3 + b_4.
%C A348517 The idea for this sequence comes from the French website Diophante (see link) where these numbers are called “tetraphile” or “4-phile”. A number that is not tetraphile is called "tetraphobe" or "4-phobe".
%C A348517 It is possible to generalize for "k-phile" or "k-phobe" numbers (see Crossrefs).
%C A348517 Some results:
%C A348517 The smallest tetraphile number is 15 = 1 + 2 + 4 + 8 and the largest tetraphobe is 48, so this sequence is infinite since every integer >= 49 is a term.
%C A348517 If m is tetraphile, q* m, q > 1, is another tetraphile number.
%C A348517 Numbers equal to 1 + 2*triphile (A160811) are tetraphile numbers, but there are other terms not of this form, as even terms.
%C A348517 There exist 23 tetraphobe numbers.
%H A348517 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).
%e A348517 As 22 = 1 + 3 + 6 + 12, 22 is a term.
%e A348517 As 33 = 1 + 2 + 6 + 24, 33 is another term.
%t A348517 Select[Range@92,Select[Select[IntegerPartitions[#,{4}],Length@Union@#==4&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]!={}&] (* _Giorgos Kalogeropoulos_, Oct 22 2021 *)
%Y A348517 k-phile numbers: A160811 \ {5} (k=3), this sequence (k=4), A348518 (k=5).
%Y A348517 k-phobe numbers: A019532 (k=3).
%K A348517 nonn
%O A348517 1,1
%A A348517 _Bernard Schott_, Oct 21 2021