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%I #11 Jun 15 2021 01:30:15
%S 1,2,4,8,10,15,16,21,32,44,45,50,52,63,64,75,99,105,117,128,130,135,
%T 136,152,153,154,165,170,182,184,189,190,195,207,230,231,232,238,250,
%U 256,266,273,290,297,310,315,322,351,375,399,405,429,434,435,441,459,484,495,512
%N Integers that are (exactly) k-deficient-perfect numbers.
%C An integer m is an exactly k-deficient-perfect number if sigma(n) = 2*n - Sum_{k} d_k, where d_i are distinct proper divisors of n.
%H Saralee Aursukaree and Prapanpong Pongsriiam, <a href="https://arxiv.org/abs/2001.06953">On Exactly 3-Deficient-Perfect Numbers</a>, arXiv:2001.06953 [math.NT], 2020.
%H FengJuan Chen, <a href="http://math.colgate.edu/~integers/t37/t37.Abstract.html">On Exactly k-deficient-perfect Numbers</a>, Integers, 19 (2019), Article A37, 1-9.
%e 117 is an exactly 2-deficient-perfect number with d1=13 and d2=39: sigma(117) = 182 = 2*117 - (13 + 39). See Theorem 1 p. 2 of Chen.
%t kdef[n_] := n == 1 || Block[{s = 2*n - DivisorSigma[1, n], d}, If[s <= 0, False, d = Most@ Divisors@ n; MemberQ[ Total /@ Subsets[d, {1, Length@ d}], s]]]; Select[ Range[512], kdef] (* _Giovanni Resta_, Jan 23 2020 *)
%o (PARI) padbin(n, len) = {my(b = binary(n)); while(length(b) < len, b = concat(0, b);); b;}
%o isok(n) = {my(d = divisors(n), s = vecsum(d))); my(nbdd = #d); for (i= 1, 2^nbdd-1, my(vecb = padbin(i, nbdd)); if (sum(j=1, nbdd, vecb[j]*d[j]) == 2*n - s, return(1)););}
%Y Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331628 (2-deficient-perfect), A331629 (3-deficient-perfect).
%K nonn
%O 1,2
%A _Michel Marcus_, Jan 23 2020
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