A356635 Numbers k that can be written as the sum of 7 divisors of k (not necessarily distinct).

7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 119, 120, 126, 128, 130

Offset: 1

Author

Wesley Ivan Hurt, Aug 18 2022

Keywords

• nonn

Comments

If k is in the sequence then so is k*m for positive m. - David A. Corneth, Aug 19 2022

Numbers that are divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138. For proof, see link. - Robert Israel, Sep 02 2022

The asymptotic density of this sequence is 17819629/37182145 = 0.479252... . - Amiram Eldar, Aug 08 2023

Examples

			10 is in the sequence since 10 = 2+2+2+1+1+1+1, where each summand divides 10.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Robert Israel, Proof that A356635 consists of all numbers divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138.

Crossrefs

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), this sequence (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

Programs

Maple

filter:= n -> ormap(t -> n mod t = 0, [7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138]):
select(filter, [$1..200]); # Robert Israel, Sep 02 2022

Mathematica

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[130], q[#, 7] &] (* Amiram Eldar, Aug 19 2022 *)

PARI

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [7,7]); \\ Michel Marcus, Aug 19 2022