cp's OEIS Frontend

A386213 Integers t having at least one nonempty subset of the set of its proper divisors for which the equation sigma(t) + r = m*t (m is any integer > 1, r is the sum of elements of such subset) is true.

%I A386213 #25 Aug 25 2025 21:30:12
%S A386213 2,4,6,8,10,12,15,16,18,20,21,24,28,30,32,36,40,42,44,45,48,50,52,54,
%T A386213 56,60,63,64,66,70,72,75,78,80,84,88,90,96,99,100,102,104,105,108,112,
%U A386213 114,117,120,126,128,130,132,135,136,138,140,144,150,152,153,154,156,160,162,165
%N A386213 Integers t having at least one nonempty subset of the set of its proper divisors for which the equation sigma(t) + r = m*t (m is any integer > 1, r is the sum of elements of such subset) is true.
%C A386213 The following table lists sequences which give k-deficient-m-perfect numbers:
%C A386213 ------------------------------------------------------------
%C A386213  k/m   |     any m     |       2       |         3         |
%C A386213 ------------------------------------------------------------
%C A386213  any k | this sequence | A331627 \ {1} |         -         |
%C A386213 ------------------------------------------------------------
%C A386213  1     | A385462       | A271816 \ {1} | A364977 \ A000396 |
%C A386213 ------------------------------------------------------------
%C A386213  2     |       -       | A331628       |         -         |
%C A386213 ------------------------------------------------------------
%C A386213  3     |       -       | A331629       |         -         |
%C A386213 ------------------------------------------------------------
%C A386213 This sequence contains all, and only, (any k)-deficient-m-perfect numbers (m = 2,3,4,...), equivalently it contains all, and only, k-deficient-(any m)-perfect numbers (k = 1,2,3,...).
%e A386213 24 is a term because for 24 the set of proper divisors is {1, 2, 3, 4, 6, 8, 12} and it has exactly 6 subsets which sum up to r satisfying the equation sigma(24) + r = k*24:
%e A386213   (1) sigma(24) + d_7(24) = 60 + 12 = 72 and 72 = 3*24,
%e A386213   (2) sigma(24) + (d_4(24) + d_6(24)) = 60 + (4 + 8) = 72 and 72 = 3*24,
%e A386213   (3) sigma(24) + (d_2(24) + d_4(24) + d_5(24)) = 60 + (2 + 4 + 6) = 72 and 72 = 3*24,
%e A386213   (4) sigma(24) + (d_1(24) + d_3(24) + d_6(24)) = 60 + (1 + 3 + 8) = 72 and 72 = 3*24,
%e A386213   (5) sigma(24) + (d_1(24) + d_2(24) + d_3(24) + d_5(24)) = 60 + (1 + 2 + 3 + 6) = 72 and 72 = 3*24,
%e A386213   (6) sigma(24) + (d_1(24) + d_2(24) + d_3(24) + d_4(24) + d_5(24) + d_6(24) + d_7(24)) = 60 + (1 + 2 + 3 + 4 + 6 + 8 + 12) = 96 and 96 = 4*24.
%e A386213 So 24 is (1, 2, 3 (in 2 variants), 4)-deficient-3-perfect and 7-deficient-4-perfect number.
%t A386213 n = 1;l={};Do[x = 1;s=DivisorSigma[1,t];A=Most[Divisors[t]];B=Subsets[A];  Do[r=Total[B[[i]]];If[Mod[s+r,t]==0,x=x+1],{i,2,2^Length[A]}];  If[x>1,AppendTo[l,t];n=n+1],{t,1,165}];l (* _James C. McMahon_, Aug 25 2025 *)
%o A386213 (Maxima)
%o A386213 (n:1, for t:1 thru 300 do (x:1, s:divsum(t), A:delete(t, divisors(t)), B:args(powerset(A)),
%o A386213               for i:2 thru 2^(length(args(A))) do (r:apply("+", args(B[i])),
%o A386213                       if mod(s+r, t)=0 then (x:x+1)),
%o A386213                                        if x>1 then (print(n, "", t), n:n+1)));
%Y A386213 Cf. A000005, A000203, A007691, A331627, A385462.
%K A386213 nonn,new
%O A386213 1,1
%A A386213 _Lechoslaw Ratajczak_, Aug 12 2025