## S000729

Numbers n such that Fibonacci(n) and Lucas(n) are abundant numbers.

18, 30, 42, 54, 90, 126, 150, 162, 198, 210, 225, 234, 270, 294, 306, 315, 330, 342, 378, 390, 414, 450, 462, 486, 510, 522, 525, 546, 558, 570, 594, 630, 666, 675, 690, 702, 714, 735, 738, 750, 774, 798, 810, 846, 855, 870, 882, 918, 930, 945, 954, 966, 990

1

The files “Fibonacci factors.txt” and “Lucas factors.txt” were created from information in the Blair Kelly website given below. A number c is called abundant if sigma(c) > 2*c, where sigma(c) is the sum of the divisors of c. It appears that 6*m is a term here, where m is a number in A056761, odd numbers n such that the cube of the number of divisors of n is greater than n.

T. D. Noe, Plot of 53 terms

Blair Kelly, Fibonacci and Lucas Factorizations

(Mma) t1 = {}; strm = OpenRead[“Fibonacci factors.txt"]; ln = -1; While[ln++; data = Read[strm, String]; data =!= EndOfFile, pos = StringPosition[data, "="][[1, 1]]; s = StringDrop[data, pos + 1]; s2 = StringSplit[s, "*"]; nums = ToExpression[s2]; sig = Times @@ DivisorSigma[1, nums]; If[sig > 2*Times @@ nums, AppendTo[t1, ln]]]; Close[strm]; t2 = {}; strm = OpenRead["Lucas factors.txt"]; ln = -1; While[ln++; data = Read[strm, String]; data =!= EndOfFile, pos = StringPosition[data, "="][[1, 1]]; s = StringDrop[data, pos + 1]; s2 = StringSplit[s, "*"]; nums = ToExpression[s2]; sig = Times @@ DivisorSigma[1, nums]; If[sig > 2*Times @@ nums, AppendTo[t2, ln]]]; Close[strm]; Intersection[t1, t2]

Cf. A074726A074316

nonn,hard

T. D. Noe, Oct 23 2015