A355641 Numbers k that can be written as the sum of 5 divisors of k (not necessarily distinct).

5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 78, 80, 81, 84, 85, 88, 90, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 120, 125, 126, 128, 130, 132, 135, 136, 138

Offset: 1

Author

Wesley Ivan Hurt, Aug 18 2022

Keywords

• nonn

Comments

Numbers that are divisible by at least one of 5, 6, 8, 9, 14 and 21. For proof see link. - Robert Israel, Sep 01 2022

The asymptotic density of this sequence is 17/35. - Amiram Eldar, Aug 08 2023

Examples

			9 is in the sequence since 9 = 3+3+1+1+1, where each summand divides 9.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Robert Israel, Proof that A355641 consists of all numbers divisible by at least one of 5, 6, 8, 9, 14, 21.

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), this sequence (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

Programs

Maple

F:= proc(x,S,j) option remember;
      local s,k;
      if j = 0  then return(x = 0) fi;
      if S = [] or x > j*S[-1]  or x < j*S[1] then return false fi;
      s:= S[-1];
      for k from 0 to min(j,floor(x/s)) do
        if procname(x-k*s, S[1..-2],j-k) then return true fi
      od;
      false
end proc:
select(t -> F(t, sort(convert(numtheory:-divisors(t),list)),5), [$1..200]); # Robert Israel, Aug 31 2022

Mathematica

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[140], q[#, 5] &] (* Amiram Eldar, Aug 19 2022 *)

PARI

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [5,5]); \\ Michel Marcus, Aug 19 2022