A386317 Integers t which satisfy 3/2 <= abundancy(t) < 2 but which are not k-deficient-perfect numbers A331627.

14, 22, 26, 34, 38, 46, 58, 62, 68, 74, 76, 82, 86, 92, 94, 98, 106, 110, 116, 118, 122, 124, 134, 142, 146, 147, 148, 158, 164, 166, 171, 172, 178, 188, 194, 202, 206, 212, 214, 218, 225, 226, 236, 242, 244, 248, 254, 255, 262, 268, 274, 278, 284, 285, 286, 292, 296, 298, 302, 314

Offset: 1

Author

Lechoslaw Ratajczak, Jul 18 2025

Keywords

• nonn

Comments

A necessary (but not sufficient) condition for an integer t to be a k-deficient-m-perfect number: (m + 1)/2 <= abundancy(t) < m:

- for m = 2: 3/2 <= abundancy(t) < 2,

- for m = 3: 2 <= abundancy(t) < 3,

- for m = 4: 5/2 <= abundancy(t) < 4.

Examples

			13 is not in this sequence because abundancy(13) = 14/13 (14/13 < 3/2).
14 is in this sequence because abundancy(14) = 12/7 (3/2 <= 12/7 < 2) but 14 is not a k-deficient-perfect number (therefore is not included in A331627).
15 is not in this sequence because abundancy(15) = 8/5 (3/2 <= 8/5 < 2) but 15 is a k-deficient-perfect number (therefore is included in A331627).
		

Crossrefs

Cf. A000203, A271816, A331627.

Programs

Maxima

(n:1, abundancy(x):=divsum(x)/x,
     for t:1 thru 500 do
        (if abundancy(t)>=3/2 and abundancy(t)<2 then
        (A:append(args(powerset(delete(t,divisors(t)))),[{0}]), b:length(A),
            for i:1 unless (divsum(t)+apply("+" , args(A[i])))/t=2 or i>=b do j:i,
               if j>=b-1 then (print(n , "" , t), n:n+1))));

PARI

isok(m) = my(d=divisors(m), ss=vecsum(d), ab=sigma(m)/m); if ((ab>=3/2) && (ab<2), d = Vec(d, #d-1); forsubset(#d, s, if (#s && (sum(i=1, #s, d[s[i]]) == 2*m - ss), return(0))); return(1)); \\ Michel Marcus, Jul 19 2025