A381060 Numbers t which are the sum of some subset of the values of k satisfying the equation (t - floor((t - k)/k)) mod k = 0 (t > 1, 1 <= k < t).

23, 29, 39, 41, 53, 59, 65, 71, 77, 79, 83, 89, 99, 101, 107, 111, 113, 119, 125, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 199, 209, 221, 227, 233, 239, 245, 251, 257, 263, 269, 279, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335, 339, 341, 349, 353, 359, 365, 371

Offset

1,1

Comments

The sequence is based on the triangle in A380305, which is a variant of the triangle in A048158. Thus, the elements of this sequence are counterparts of pseudoperfect numbers (A005835), such as the elements of A375595 are counterparts of abundant numbers (A005101).

The sequence includes all elements of A380153.

The first even element of this sequence is a(768) = 4094.

Links

Table of n, a(n) for n=1..59.

Example

23 is in this sequence because the only k's < 23 satisfying the equation (23 - floor((23 - k)/k)) mod k = 0 are: 1, 5, 7, 11, hence: 5+7+11 = 23.
29 is in this sequence because the only k's < 29 satisfying the equation (29 - floor((29 - k)/k)) mod k = 0 are: 1, 2, 3, 5, 9, 14, hence: 1+2+3+9+14 = 29 and 1+5+9+14 = 29.
47 is not in this sequence because the only k's < 47 satisfying the equation (47 - floor((47 - k)/k)) mod k = 0  are: 1, 3, 7, 11, 15, 23 and no subset of these numbers adds to 47.

Prog

(Maxima)
(kill(all), s(y):=(f(i, j):=mod(i-floor((i-j)/j), j), s:0, x:1,
      for k:1 thru floor(y/2) do
              (if f(y, k)=0 then
              (s:s+k, B[x]:k, x:x+1)),
B:setify(makelist(B[r], r, 1, x-1)), s),
n:1, for t:2 thru 1000 do
              (if s(t)>=t  then
                     (for b:2 while b<=x-1 and e#t do
                     (C:args(powerset(B, b)),
                            for h:1 while h<=length(C) and e#t do
                            (e:apply("+" , args(C[h])),
                                    if e=t then
                                    (print(n , " " , t), n:n+1))))));

Crossrefs

Cf. A005835, A048158, A375595, A380153, A380305.

Sequence in context: A101802 A363074 A156983 * A230456 A174196 A144226

Adjacent sequences: A381057 A381058 A381059 * A381061 A381062 A381063

Keyword

nonn

Author

Lechoslaw Ratajczak, Feb 12 2025

Status

approved