A348518 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.

31, 39, 43, 45, 46, 47, 55, 57, 58, 59, 61, 62, 63, 64, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 99, 100, 101, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Author

Bernard Schott, Oct 21 2021

Keywords

• nonn

Comments

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”.

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

Some results:

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.

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

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

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

There exist 68 pentaphobe numbers.

Examples

			As 43 = 1 + 2 + 4 + 12 + 24, 43 is a term.
As 89 = 1 + 4 + 12 + 24 + 48, 89 is another term.
		

Links

• Diophante, A496 - Pentaphiles et pentaphobes (in French).

Crossrefs

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

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

Programs

Mathematica

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