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A331627
Integers that are (exactly) k-deficient-perfect numbers.
2
1, 2, 4, 8, 10, 15, 16, 21, 32, 44, 45, 50, 52, 63, 64, 75, 99, 105, 117, 128, 130, 135, 136, 152, 153, 154, 165, 170, 182, 184, 189, 190, 195, 207, 230, 231, 232, 238, 250, 256, 266, 273, 290, 297, 310, 315, 322, 351, 375, 399, 405, 429, 434, 435, 441, 459, 484, 495, 512
OFFSET
1,2
COMMENTS
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.
LINKS
Saralee Aursukaree and Prapanpong Pongsriiam, On Exactly 3-Deficient-Perfect Numbers, arXiv:2001.06953 [math.NT], 2020.
FengJuan Chen, On Exactly k-deficient-perfect Numbers, Integers, 19 (2019), Article A37, 1-9.
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(PARI) padbin(n, len) = {my(b = binary(n)); while(length(b) < len, b = concat(0, b); ); b; }
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)); ); }
CROSSREFS
Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331628 (2-deficient-perfect), A331629 (3-deficient-perfect).
Sequence in context: A178215 A067925 A323102 * A102431 A076919 A336659
KEYWORD
nonn
AUTHOR
Michel Marcus, Jan 23 2020
STATUS
approved