A331627 Integers that are (exactly) k-deficient-perfect numbers.

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

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).

Sequence in context: A178215 A067925 A323102 * A102431 A076919 A336659

Adjacent sequences: A331624 A331625 A331626 * A331628 A331629 A331630

Keyword

nonn

Author

Michel Marcus, Jan 23 2020

Status

approved