A295265 Numbers m such that sum of its i first divisors equals the sum of its j first non-divisors for some i, j.

4, 8, 10, 13, 14, 16, 19, 20, 21, 22, 26, 28, 30, 32, 34, 38, 39, 40, 43, 44, 46, 50, 52, 53, 56, 58, 60, 62, 63, 64, 68, 70, 72, 74, 76, 80, 82, 86, 88, 89, 90, 92, 94, 98, 99, 100, 103, 104, 106, 110, 111, 112, 116, 117, 118, 122, 124, 128, 130, 132, 134, 135

Offset

1,1

Comments

Or numbers m such that Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) for some i, j where d(k) are the i first divisors and nd(k) the j non-divisors of m.

The corresponding sums are 3, 3, 3, 14, 3, 3, 20, 3, 11, 3, 3, (3 or 14), 11, 3, 3, 3, 17, 3, 44, 3, 3, 3, 3, 54, 3, 3, 15, 3, 11, 3, 3, 3, 33, 3, 3, 3, ... containing the set of primes {3, 11, 17, 23, 29, 37, 41, 43, 53, 59, 61, 71, 79, ...}.

The equality Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) is not always unique, for instance for a(12) = 28, we find 1 + 2 = 3 and 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14.

The primes of the sequence are 13, 19, 43, 53, 89, 103, 151, 229, 251, 349, 433, ... (primes of the form k(k+1)/2 - 2; see A124199).

+-----+-----+-----+------+-----------------------------------------+

| n | i | j | a(n) | Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) |

+-----+-----+-----+------+-----------------------------------------+

| 1 | 2 | 1 | 4 | 1 + 2 = 3 |

| 2 | 2 | 1 | 8 | 1 + 2 = 3 |

| 3 | 2 | 1 | 10 | 1 + 2 = 3 |

| 4 | 2 | 4 | 13 | 1 + 13 = 2 + 3 + 4 + 5 = 14 |

| 5 | 2 | 1 | 14 | 1 + 2 = 3 |

| 6 | 2 | 1 | 16 | 1 + 2 = 3 |

| 7 | 2 | 5 | 19 | 1 + 19 = 2 + 3 + 4 + 5 + 6 = 20 |

| 8 | 2 | 1 | 20 | 1 + 2 = 3 |

| 9 | 3 | 3 | 21 | 1 + 3 + 7 = 2 + 4 + 5 = 11 |

| 10 | 2 | 1 | 22 | 1 + 2 = 3 |

| 11 | 2 | 1 | 26 | 1 + 2 = 3 |

| 12 | 2 | 1 | 28 | 1 + 2 = 3 |

| | 4 | 3 | 28 | 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14 |

| 13 | 4 | 2 | 30 | 1 + 2 + 3 + 5 = 4 + 7 = 11 |

| 14 | 2 | 1 | 32 | 1 + 2 = 3 | (End)

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

30 is in the sequence because d(1) + d(2) + d(3) + d(4) = 1 + 2 + 3 + 5 = 11 and nd(1) + nd(2) = 4 + 7 = 11.

Maple

with(numtheory):nn:=300:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):lst:={}:ii:=0:
  for i from 1 to n do:
   lst:=lst union {i}:
  od:
    lst:=lst minus d:n1:=nops(lst):
     for m from 1 to n0 while(ii=0) do:
      s1:=sum(‘d[i]’, ‘i’=1..m):
       for j from 1 to n1 while(ii=0) do:
        s2:=sum(‘lst[i]’, ‘i’=1..j):
         if s1=s2
          then
          ii:=1:printf(`%d, `, n):
         else
         fi:
        od:
     od:
  od:

Mathematica

fQ[n_] := Block[{d = Divisors@ n}, nd = nd = Complement[Range@ n, d]; Intersection[Accumulate@ d, Accumulate@ nd] != {}]; Select[ Range@135, fQ] (* Robert G. Wilson v, Mar 06 2018 *)

Prog

(PARI) isok(n) = {d = divisors(n); psd = vector(#d, k, sum(j=1, k, d[j])); nd = setminus([1..n], d); psnd = vector(#nd, k, sum(j=1, k, nd[j])); #setintersect(psd, psnd) != 0; } \\ Michel Marcus, May 05 2018

Crossrefs

Cf. A064510, A124199, A185729, A240698.

Sequence in context: A238749 A310984 A346002 * A172153 A310985 A342734

Adjacent sequences: A295262 A295263 A295264 * A295266 A295267 A295268

Keyword

nonn

Author

Michel Lagneau, Feb 22 2018

Status

approved