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

1,1

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

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.

Example

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

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 *)

Prog

(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

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).

Sequence in context: A296562 A057854 A033266 * A363701 A353193 A102408

Adjacent sequences: A355638 A355639 A355640 * A355642 A355643 A355644

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 18 2022

Status

approved