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%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
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