This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354591 #45 Jul 17 2025 07:36:19
%S A354591 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 A354591 68,70,72,76,78,80,84,88,90,92,96,100,102,104,108,110,112,114,116,120,
%U A354591 124,126,128,130,132,136,138,140,144,148,150,152,156,160,162,164,168,170,172
%N A354591 Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).
%C A354591 All terms are even. - _Robert Israel_, Aug 31 2022
%C A354591 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 A354591 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 A354591 The asymptotic density of this sequence is 11/30. - _Amiram Eldar_, Aug 08 2023
%H A354591 Robert Israel, <a href="/A354591/b354591.txt">Table of n, a(n) for n = 1..10000</a>
%H A354591 <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).
%F A354591 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 2*a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - 2*a(n-16) + 2*a(n-17) - 2*a(n-18) + 2*a(n-19) - 2*a(n-20) + 2*a(n-21) - a(n-22). - _Wesley Ivan Hurt_, Jun 29 2024
%F A354591 G.f.: 2*x*(2 - x + 2*x^2 - x^3 + 2*x^4 + x^6 + 2*x^8 + x^10 + 2*x^12 + x^14 + 2*x^16 - x^17 + 2*x^18 - x^19 + 2*x^20)/((x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(1 + x + x^5 + x^6 + x^7 + x^8 + x^9 + x^2 + x^4 + x^3 + x^10)*(x - 1)^2). - _Wesley Ivan Hurt_, Jul 17 2025
%e A354591 20 is in the sequence since 20 = 10+5+4+1 = 5+5+5+5 where each summand divides 20.
%p A354591 F:= proc(x,S,j) option remember;
%p A354591 local s,k;
%p A354591 if j = 0 then return(x = 0) fi;
%p A354591 if S = [] or x > j*S[-1] then return false fi;
%p A354591 s:= S[-1];
%p A354591 for k from 0 to min(j,floor(x/s)) do
%p A354591 if procname(x-k*s, S[1..-2],j-k) then return true fi
%p A354591 od;
%p A354591 false
%p A354591 end proc:
%p A354591 select(t -> F(t, sort(convert(numtheory:-divisors(t),list)),4), [$1..200]); # _Robert Israel_, Aug 31 2022
%t A354591 q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[200], q[#, 4] &] (* _Amiram Eldar_, Aug 19 2022 *)
%t A354591 CoefficientList[Series[2 (2 - x + 2*x^2 - x^3 + 2*x^4 + x^6 + 2*x^8 + x^10 + 2*x^12 + x^14 + 2*x^16 - x^17 + 2*x^18 - x^19 + 2*x^20)/((x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(1 + x + x^5 + x^6 + x^7 + x^8 + x^9 + x^2 + x^4 + x^3 + x^10)*(x - 1)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jul 17 2025 *)
%o A354591 (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 A354591 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 A354591 Cf. A080671.
%K A354591 nonn
%O A354591 1,1
%A A354591 _Wesley Ivan Hurt_, Aug 18 2022