OFFSET
1,1
COMMENTS
Subsequence of A111592.
The corresponding Fibonacci numbers are given by the sequence {b(n)} = 2, 1, 5, 2, 2, 1, 1, 1, 8, 2, 1, 2, 2, 8, 8, 3, 21, 8, 34, 8, 21, 8, 2, 3, 13, 1, 3, 2, 1, ....
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..46
Terry Trotter, Admirable Numbers, 2009. [Wayback Machine copy] [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - N. J. A. Sloane, Mar 29 2018]
EXAMPLE
40 is in the sequence because sigma(40) - 2*5 = 90 - 10 = 80 = 2*40, where 5 is a Fibonacci number, or 1 + 2 + 4 + 8 + 10 + 20 - 5 = 40 where the subtracted divisor is 5.
MAPLE
with(numtheory):
for n from 1 to 20000 do:
x:=divisors(n):n0:=nops(x):
for i from 1 to n0 do:
u:=sqrt(5*x[i]^2-4):v:=sqrt(5*x[i]^2+4):
if (floor(u)=u or floor(v)=v) and sigma(n)-2*x[i]=2*n
then
printf(`%d %d \n`, n, x[i]):
else
fi:
od:
od:
MATHEMATICA
With[{nn = 10^6}, Function[s, Flatten@ Position[#, 1] &@ Table[Total@ Boole@ Map[MemberQ[s, #] &, Select[Most@ Divisors@ n, Function[d, DivisorSigma[1, n] - 2 d == 2 n]]], {n, nn}]]@ Fibonacci@ Range[2 + Floor@ Log[GoldenRatio, nn]]] (* Michael De Vlieger, Feb 24 2017 *) (* or *)
fibQ[n_] := IntegerQ@ Sqrt[5 n^2 + 4] || IntegerQ@ Sqrt[5 n^2 - 4]; ok[n_] := Block[{d = DivisorSigma[1, n] - 2 n}, d>0 && EvenQ@d && Mod[n, d/2] == 0 && fibQ[d/2]]; Select[Range[10^6], ok] (* faster, Giovanni Resta, Mar 10 2017 *)
PROG
(PARI) isadmirable(n)=if(issquare(n)||issquare(n/2), 0, my(d=sigma(n)/2-n); (d>0 && d!=n && n%d==0)*d);
isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8))
isok(n) = (d=isadmirable(n)) && isfib(d); \\ Michel Marcus, Mar 10 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 21 2017
EXTENSIONS
More terms from Michel Marcus, Mar 10 2017
STATUS
approved