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

Author

Michel Lagneau, Aug 07 2017

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A008776, A025192.

Programs

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 *)

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

Extensions

Name corrected by Jon E. Schoenfield, Sep 11 2017