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

Author

Michel Lagneau, Feb 22 2018

Keywords

• nonn

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 |

Examples

			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.
		

Links

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

Crossrefs

Cf. A064510, A124199, A185729, A240698.

Programs

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 *)

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