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A356660 Numbers k that can be written as the sum of 10 divisors of k (not necessarily distinct).

%I A356660 #24 Oct 09 2022 09:42:22
%S A356660 10,12,14,16,18,20,22,24,26,28,30,32,34,36,40,42,44,48,50,52,54,56,60,
%T A356660 64,66,68,70,72,76,78,80,84,88,90,92,96,98,100,102,104,108,110,112,
%U A356660 114,116,120,126,128,130,132,136,138,140,144,150,152,154,156,160,162
%N A356660 Numbers k that can be written as the sum of 10 divisors of k (not necessarily distinct).
%C A356660 From _David A. Corneth_, Oct 08 2022: (Start)
%C A356660 All terms are even. Proof: suppose a term is odd. Then all divisors are odd. Adding 10 odd numbers gives an even number. A contradiction.
%C A356660 If k is a term then so is k*m for m >= 1. Proof: Multiply each divisor in this sum of 10 divisors that give k with m. Then each term is a divisor of k*m and their sum is k*m. (End)
%H A356660 David A. Corneth, <a href="/A356660/b356660.txt">Table of n, a(n) for n = 1..10000</a>
%e A356660 14 is in the sequence since 14 = 2+2+2+2+1+1+1+1+1+1, where each summand divides 14.
%o A356660 (Python)
%o A356660 from sympy import divisors
%o A356660 def t_sum_of_n_div(n, target):
%o A356660     out, p = [], divisors(n)[::-1][1:]
%o A356660     def dfs(t, divs,  index_s, kk):
%o A356660         if len(out)!=0 or kk>target:return
%o A356660         if kk == target and t == 0:
%o A356660             out.append(divs)
%o A356660             return
%o A356660         for i in range(index_s, len(p)):
%o A356660             if t >= p[i]:
%o A356660                 temp_divs = divs.copy()
%o A356660                 temp_divs.append(p[i])
%o A356660                 dfs(t-p[i], temp_divs, i, kk+1)
%o A356660     dfs(n, [], 0, 0)
%o A356660     return out
%o A356660 terms = [i for i in range(2, 200) if len(t_sum_of_n_div(i,10))!=0]
%o A356660 print(terms) # _Gleb Ivanov_, Sep 02 2022
%o A356660 (PARI) upto(n) = { my(v = vector(n,i,-1), t = 0); forstep(i = 2, n, 2, if(v[i] == -1, v[i] = is(i); if(v[i] == 1, for(j = 2, n \ i, v[i*j] = 1; ) ) ); ); select(x->x >= 1, v, 1); }
%o A356660 is(n, {qd = 10}) = { my(d = divisors(n), res = 0); d = d[^#d]; forvec(x = vector(qd-1, i, [1, #d]), s = sum(i = 1, qd-1, d[x[i]]); if(n - s >= d[x[qd - 1]], if(n % (n - s) == 0,  return(1); ) ) , 1 ); 0 } \\ _David A. Corneth_, Oct 08 2022
%Y A356660 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), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), this sequence (j=10).
%K A356660 nonn
%O A356660 1,1
%A A356660 _Wesley Ivan Hurt_, Aug 20 2022