A355641 - OEIS

%I #32 Aug 08 2023 03:22:14

%S 5,6,8,9,10,12,14,15,16,18,20,21,24,25,27,28,30,32,35,36,40,42,45,48,

%T 50,54,55,56,60,63,64,65,66,70,72,75,78,80,81,84,85,88,90,95,96,98,99,

%U 100,102,104,105,108,110,112,114,115,117,120,125,126,128,130,132,135,136,138

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

%C 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

%C The asymptotic density of this sequence is 17/35. - _Amiram Eldar_, Aug 08 2023

%H Robert Israel, <a href="/A355641/b355641.txt">Table of n, a(n) for n = 1..10000</a>

%H Robert Israel, <a href="/A355641/a355641.pdf">Proof that A355641 consists of all numbers divisible by at least one of 5, 6, 8, 9, 14, 21</a>.

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

%p F:= proc(x,S,j) option remember;

%p       local s,k;

%p       if j = 0  then return(x = 0) fi;

%p       if S = [] or x > j*S[-1]  or x < j*S[1] then return false fi;

%p       s:= S[-1];

%p       for k from 0 to min(j,floor(x/s)) do

%p         if procname(x-k*s, S[1..-2],j-k) then return true fi

%p       od;

%p       false

%p end proc:

%p select(t -> F(t, sort(convert(numtheory:-divisors(t),list)),5), [$1..200]); # _Robert Israel_, Aug 31 2022

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

%o (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

%Y 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).

%K nonn

%O 1,1

%A _Wesley Ivan Hurt_, Aug 18 2022