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

Author

Michel Marcus, Jan 23 2020

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331628 (2-deficient-perfect), A331629 (3-deficient-perfect), A386317.

Programs

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 *)

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