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.

15, 19, 21, 22, 23, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

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 “tetraphile” or “4-phile”. A number that is not tetraphile is called "tetraphobe" or "4-phobe".

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

Some results:

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.

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

Numbers equal to 1 + 2*triphile (A160811) are tetraphile numbers, but there are other terms not of this form, as even terms.

There exist 23 tetraphobe numbers.

Examples

			As 22 = 1 + 3 + 6 + 12, 22 is a term.
As 33 = 1 + 2 + 6 + 24, 33 is another term.
		

Links

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

Crossrefs

k-phile numbers: A160811 \ {5} (k=3), this sequence (k=4), A348518 (k=5).

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

Programs

Mathematica

Select[Range@92,Select[Select[IntegerPartitions[#,{4}],Length@Union@#==4&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]!={}&] (* Giorgos Kalogeropoulos, Oct 22 2021 *)