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%I #16 Jun 17 2016 00:44:36
%S 2,14,18,230,238,4958,53430,57930,64506,65586,68226,70730,77270,78638,
%T 81926,84986,88826,90446,91306,1006350,1248054,1341950,18177726,
%U 19033854,19603430,21044030,22356798,22395522,22876730,23954170,24241966,24840710,24883910,25285666,25306246
%N Even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is 2F.
%C 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)).
%C 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
%C 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.
%e 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.
%p with(numtheory):
%p for n from 2 by 2 to 10^7 do:
%p y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
%p for k from 1 to n1 do:
%p if irem(y[k], 2)=0
%p then
%p s0:=s0+ y[k]:
%p else
%p s1:=s1+ y[k]:
%p fi:
%p od:
%p if s0=2*s1
%p then
%p ii:=0:
%p x:=sqrt(5*s1^2+4):y:=sqrt(5*s1^2-4):
%p if x=floor(x) or y=floor(y)
%p then
%p printf ( "%d %d \n",n,s1):
%p else
%p fi:
%p fi:
%p od:
%t 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 *)
%o (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
%Y Cf. A000045, A000918, A016825, A087943.
%K nonn
%O 1,1
%A _Michel Lagneau_, Jun 09 2016
%E a(23)-a(35) from _Michel Marcus_, Jun 14 2016
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