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URL: https://oeis.org/A331627

⇱ A331627 - OEIS

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A331627
Integers that are (exactly) k-deficient-perfect numbers.
5
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 it has k distinct proper divisors d_i such that sigma(m) = 2*m - Sum_{i=1..k} d_i.
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) isok(m) = my(d=divisors(m), ss=sigma(m)); d = Vec(d, #d-1); forsubset(#d, s, if (#s && (sum(i=1, #s, d[s[i]]) == 2*m - ss), return(1)); ); \\ Michel Marcus, Dec 29 2024
CROSSREFS
Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331628 (2-deficient-perfect), A331629 (3-deficient-perfect), A386317.
Sequence in context: A178215 A067925 A323102 * A102431 A076919 A336659
KEYWORD
nonn
AUTHOR
Michel Marcus, Jan 23 2020
STATUS
approved