A282754 Admirable numbers such that the subtracted divisor is a Fibonacci number.

12, 20, 40, 70, 88, 104, 464, 650, 1504, 1888, 1952, 4030, 5830, 7192, 7912, 8925, 9555, 10792, 13736, 17272, 30555, 30592, 32128, 32445, 78975, 130304, 442365, 521728, 522752, 1713592, 1848964, 4526272, 8353792, 8378368, 8382464, 9928792, 11547352, 17999992

Offset: 1

Author

Michel Lagneau, Feb 21 2017

Keywords

• nonn

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, ....

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..46
• Terry Trotter, Admirable Numbers, 2009. [Wayback Machine copy]

Crossrefs

Cf. A000045, A010056, A111592.

Programs

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 *)

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

Extensions

More terms from Michel Marcus, Mar 10 2017