A290582 Numbers m > 2 such that every divisor > 2 is the sum of two or more consecutive divisors.

6, 18, 54, 66, 162, 486, 726, 1458, 4374, 7986, 13122, 39366, 87846, 118098, 354294, 530226, 966306, 1062882, 3188646, 9565938, 10629366, 28697814, 43035786, 86093442, 116923026, 258280326, 578476566, 774840978, 1286153286, 2324522934, 6973568802, 14147686146

Offset

1,1

Comments

a(n) is even because the divisors are {d(1), d(2), d(3), ...} with d(1) = 1, and if d(2) is odd then d(3) = d(2) + 1 is even, a contradiction. Therefore d(2) = 2 and d(3) = 3.

a(n) == 0 (mod 6).

The sequence is infinite because the numbers of the form 2*3^i (A025192) and the numbers of the form 6*11^i for i >= 1 are in the sequence.

The divisors of 2*3^i are {d(1), d(2), d(3), ...} = {1, 2, 3, 6, 9, 18, 27, 54, ...} where d(1 + 2i) = 3^i for i >= 0 and d(2i) = 2*3^(i-1) for i >= 1.

The divisors of the form 6*11^k are {d(1), d(2), d(3), ...} = {1, 2, 3, 6, 11, 22, 33, 66, 121, ...} where d(i + 4j) = i*11^j for i >= 1 and j >= 0 and where d(4j) = 6*11^(j-1) for j >= 1.

No term can be divisible by 4, 5, 7, 9, 13, 17, or 19. Up to 5*10^11, the only terms which are divisible by a prime > 11 are 530226 = 2*3^5*1091, 43035786 = 2*3^7*9839, 578476566 = 2*3^5*1091^2, and 2*3^7*9839^2. Larger such terms are 2*3*11^12*6904542428779 and 2*3^29*308836698141971. - Giovanni Resta, Aug 07 2017

Links

Table of n, a(n) for n=1..32.

Example

66 is in the sequence because the divisors are {1, 2, 3, 6, 11, 22, 33, 66} and:
3 = 2 + 1;
6 = 3 + 2 + 1;
11 = 6 + 3 + 2;
22 = 11 + 6 + 3 + 2;
33 = 22 + 11;
66 = 33 + 22 + 11.

Maple

with(numtheory):nn:=10^5:
for n from 4 to nn do:
  d:=divisors(n):n1:=nops(d):it:=0:
   for k from 3 to n1 do:
     for j from 1 to k-1 do:
       s:=sum('d[k-i]', 'i'=1..j):
        if s=d[k]
         then
         it:=it+1:
         else
        fi:
     od:
    od:
     if n1>2 and it = n1-2
      then
      printf(`%d, `, n):
      else
     fi:
    od:

Mathematica

Select[Range[3, 10^5], Function[d, Function[t, AllTrue[ TakeWhile[ Reverse@ d, # > 2 &], MemberQ[t, #] &]]@ Union@ Flatten@ Array[Total /@ Partition[d, #, 1] &, Length@ d - 1, 2]]@ Divisors@ # &] (* Michael De Vlieger, Aug 07 2017 *)

Prog

(PARI) isokds(k, v) = {vsmall = select(x->(x < k), v); for (i=1, #vsmall, s = v[i]; if (s > k, break); for (j=i+1, #vsmall, s += vsmall[j]; if (s > k, break, if (k == s, return(1))); ); ); return (0); }
isok(n) = {if (n>2, my(d = divisors(n)); for (i=1, #d, if (d[i] > 2, if (! isokds(d[i], d), return (0)); ); ); return(1); )} \\ Michel Marcus, Aug 07 2017

Crossrefs

Cf. A008776, A025192.

Sequence in context: A001216 A318484 A079843 * A346071 A076941 A357198

Adjacent sequences: A290579 A290580 A290581 * A290583 A290584 A290585

Keyword

nonn

Author

Michel Lagneau, Aug 07 2017

Extensions

Name corrected by Jon E. Schoenfield, Sep 11 2017

Status

approved