A354591 Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).

4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 170, 172

Offset: 1

Author

Wesley Ivan Hurt, Aug 18 2022

Keywords

• nonn

Comments

All terms are even. - Robert Israel, Aug 31 2022

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)

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

The asymptotic density of this sequence is 11/30. - Amiram Eldar, Aug 08 2023

Examples

			20 is in the sequence since 20 = 10+5+4+1 = 5+5+5+5 where each summand divides 20.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).

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), this sequence (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

Cf. A080671.

Programs

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] 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)),4), [$1..200]); # Robert Israel, Aug 31 2022

Mathematica

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[200], q[#, 4] &] (* Amiram Eldar, Aug 19 2022 *)
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 *)

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

Formula

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

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