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%I #41 Nov 10 2023 10:43:17
%S 4,6,8,10,12,16,18,20,24,28,30,32,36,40,42,44,48,50,52,54,56,60,64,66,
%T 68,70,72,76,78,80,84,88,90,92,96,100,102,104,108,110,112,114,116,120,
%U 124,126,128,130,132,136,138,140,144,148,150,152,156,160,162,164,168,170,172
%N Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).
%C All terms are even. - _Robert Israel_, Aug 31 2022
%C Is it true that a(n) = 2*A080671(n)? - _Michel Marcus_, Sep 01 2022 (True for n <= 10000. - _N. J. A. Sloane_, Sep 01 2022)
%C This is true. In other words, k is in the sequence if and only if k is even and divisible by 3, 4 or 5. Proof: the positive integer solutions of 1/a + 1/b + 1/c + 1/d = 1 can be enumerated explicitly, and each contains at least one even number and at least one divisible by 3, 4 or 5. Of course k = k/a + k/b + k/c + k/d if and only if 1 = 1/a + 1/b + 1/c + 1/d, and this writes k as the sum of 4 divisors of k if k is divisible by a,b,c, and d. If k is even and divisible by 3, we can use 1 = 1/3 + 1/3 + 1/6 + 1/6; if divisible by 4, 1 = 1/4 + 1/4 + 1/4 + 1/4; if even and divisible by 5, 1 = 1/2 + 1/5 + 1/5 + 1/10. - _Robert Israel_, Sep 01 2022
%C The asymptotic density of this sequence is 11/30. - _Amiram Eldar_, Aug 08 2023
%H Robert Israel, <a href="/A354591/b354591.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_22">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).
%e 20 is in the sequence since 20 = 10+5+4+1 = 5+5+5+5 where each summand divides 20.
%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] 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)),4), [$1..200]); # _Robert Israel_, Aug 31 2022
%t q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[200], q[#, 4] &] (* _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)), , [4,4]); \\ _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), this sequence (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).
%Y Cf. A080671.
%K nonn
%O 1,1
%A _Wesley Ivan Hurt_, Aug 18 2022
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