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A274062
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Even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is 2F.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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