A330777 Numbers k such that k and Lucas(k) have the same number of divisors.

1, 2, 5, 7, 10, 11, 13, 14, 17, 19, 24, 26, 28, 31, 37, 38, 40, 41, 47, 53, 61, 62, 71, 79, 86, 88, 113, 152, 178, 202, 248, 313, 353, 458, 488, 503, 586, 613, 617, 856, 863, 914, 1082, 1097, 1306, 1318, 1361, 1784

Offset: 1

Author

Chai Wah Wu, Dec 31 2019

Keywords

• nonn
• more

Comments

All prime terms of A001606 (i.e., terms in A001606 that are not nontrivial powers of 2) are terms of this sequence.

Conjecture: all terms are of the form 2^k*p for k >= 0 and p prime.

It is unknown whether 1816 is a term (the smallest number for which membership in the sequence is unknown); it depends on whether Lucas(1816)/47 is a semiprime or not. The following composite numbers are terms of the sequence: 3106, 3928, 4006, 5414, 5498, 14318, 20578. - Chai Wah Wu, Jan 03 2020

Links

• Blair Kelly, Fibonacci and Lucas Factorizations.

Crossrefs

Cf. A000032, A001606, A080651.

Programs

Mathematica

Select[Range[100],DivisorSigma[0,#]==DivisorSigma[0,LucasL[#]]&] (* Metin Sariyar, Jan 03 2020 *)

PARI

for(k=1,320,if(numdiv(k)==numdiv(fibonacci(k+1)+fibonacci(k-1)),print1(k,", "))) \\ Hugo Pfoertner, Jan 03 2020