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

1,1

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

Cf. A000045, A010056, A111592.

Sequence in context: A260905 A211415 A209973 * A063690 A229355 A259174

Adjacent sequences: A282751 A282752 A282753 * A282755 A282756 A282757

Keyword

nonn

Author

Michel Lagneau, Feb 21 2017

Extensions

More terms from Michel Marcus, Mar 10 2017

Status

approved