A274062 Even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is 2F.

2, 14, 18, 230, 238, 4958, 53430, 57930, 64506, 65586, 68226, 70730, 77270, 78638, 81926, 84986, 88826, 90446, 91306, 1006350, 1248054, 1341950, 18177726, 19033854, 19603430, 21044030, 22356798, 22395522, 22876730, 23954170, 24241966, 24840710, 24883910, 25285666, 25306246

Offset: 1

Author

Michel Lagneau, Jun 09 2016

Keywords

• nonn

Comments

a(n) == 2,6,10 (mod 12) i.e. a(n) == 2 (mod 4) so this sequence is a subsequence of A016825 (of which 3|sigma(A016825(n))).

The corresponding Fibonacci numbers F are 1, 8, 13, 144, 144, 2584, 46368, 46368, 46368, 46368,... with index 1 (or 2), 6, 7, 12, 12, 18, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 30, 30, 30.

The sequence is generalizable with the following definition: even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is (2^k -2)*F = A000918(k)*F with k>1. The corresponding sequences b(n,k) are of the form b(n,k) = a(n)*2^(k-2) where a(n) is the primitive sequence.

Examples

			18 is in the sequence because: its divisors are {1, 2, 3, 6, 9, 18}; the sum of its odd divisors is 1 + 3 + 9 = 13, a Fibonacci number, and the sum of its even divisors is 2 + 6 + 18 = 26 = 2*13.
		

Crossrefs

Cf. A000045, A000918, A016825, A087943.

Programs

Maple

with(numtheory):
for n from 2 by 2  to 10^7 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     if s0=2*s1
      then
      ii:=0:
        x:=sqrt(5*s1^2+4):y:=sqrt(5*s1^2-4):
         if x=floor(x) or y=floor(y)
          then
          printf ( "%d %d \n",n,s1):
           else
          fi:
        fi:
     od:

Mathematica

t = Fibonacci@ Range@ 40; Select[Range[2, 2*10^6, 4], Function[d, And[Total@ Select[d, EvenQ] == 2 #, MemberQ[t, #]] &@ Total@ Select[d, OddQ]]@ Divisors@ # &] (* Michael De Vlieger, Jun 09 2016 *)

PARI

isok(n) = sod = sumdiv(n, d, d*(d % 2)); (2*sod == sumdiv(n, d, d*(1-(d % 2)))) && (issquare(5*sod^2-4) || issquare(5*sod^2+4)); \\ Michel Marcus, Jun 09 2016

Extensions

a(23)-a(35) from Michel Marcus, Jun 14 2016