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

1,1

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

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.

Example

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

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 *)

Prog

(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

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).

Sequence in context: A065976 A236683 A120200 * A343297 A106108 A120309

Adjacent sequences: A356632 A356633 A356634 * A356636 A356637 A356638

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 18 2022

Status

approved