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

1,1

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.

Links

Table of n, a(n) for n=1..60.

Example

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).

Prog

(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

Crossrefs

Cf. A000203, A271816, A331627.

Sequence in context: A113190 A169804 A354811 * A392985 A092112 A306146

Adjacent sequences: A386314 A386315 A386316 * A386318 A386319 A386320

Keyword

nonn

Author

Lechoslaw Ratajczak, Jul 18 2025

Status

approved