A348518 - OEIS

%I #16 Apr 03 2023 15:39:34

%S 31,39,43,45,46,47,55,57,58,59,61,62,63,64,67,69,70,71,73,75,76,77,78,

%T 79,81,82,83,85,86,87,88,89,90,91,92,93,94,95,96,99,100,101,103,105,

%U 106,107,109,110,111,112,113,114,115,116,117,118,119,121,122,123,124,125,126,127,128

%N Positive integers m with the property that there are 5 positive integers b_1 < b_2 < b_3 < b_4 < b_5 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, b_4 divides b_5, and m = b_1 + b_2 + b_3 + b_4 + b_5.

%C The idea for this sequence comes from the French website Diophante (see link) where these numbers are called “pentaphile” or “5-phile”. A number that is not pentaphile is called “pentaphobe” or “5-phobe”.

%C It is possible to generalize for “k-phile” or “k-phobe” numbers (see Crossrefs).

%C Some results:

%C The smallest pentaphile number is 31 = 1 + 2 + 4 + 8 + 16 and the largest pentaphobe number is 240, so, this sequence is infinite since all integers >= 241 are terms.

%C Every term m = r * (1+s*t) with r > 0, s > 1 and t is a tetraphile number (A348517).

%C Odd numbers equal to 1 + 2*t where t is tetraphile (A348517) are pentaphile numbers, so odd numbers >= 99 are pentaphile.

%C If m is pentaphile, q* m, q > 1, is another pentaphile number.

%C There exist 68 pentaphobe numbers.

%H 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 As 43 = 1 + 2 + 4 + 12 + 24, 43 is a term.

%e As 89 = 1 + 4 + 12 + 24 + 48, 89 is another term.

%t Select[Range@100,Select[Select[IntegerPartitions[#,{5}],Length@Union@#==5&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]!={}&] (* _Giorgos Kalogeropoulos_, Oct 22 2021 *)

%Y k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), this sequence (k=5).

%Y k-phobe numbers: A019532 (k=3).

%K nonn

%O 1,1

%A _Bernard Schott_, Oct 21 2021