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A330777 Numbers n such that n and Lucas(n) have the same number of divisors. 0


%S 1,2,5,7,10,11,13,14,17,19,24,26,28,31,37,38,40,41,47,53,61,62,71,79,

%T 86,88,113,152,178,202,248,313,353,458,488,503,586,613,617,856,863,

%U 914,1082,1097,1306,1318,1361,1784

%N Numbers n such that n and Lucas(n) have the same number of divisors.

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

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

%C 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

%H Blair Kelly, <a href="http://mersennus.net/fibonacci/">Fibonacci and Lucas Factorizations</a>.

%t Select[Range[100],DivisorSigma[0,#]==DivisorSigma[0,LucasL[#]]&] (* _Metin Sariyar_, Jan 03 2020 *)

%o (PARI) for(k=1,320,if(numdiv(k)==numdiv(fibonacci(k+1)+fibonacci(k-1)),print1(k,", "))) \\ _Hugo Pfoertner_, Jan 03 2020

%Y Cf. A000032, A001606, A080651.

%K nonn,more

%O 1,2

%A _Chai Wah Wu_, Dec 31 2019

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Last modified October 1 13:04 EDT 2020. Contains 337443 sequences. (Running on oeis4.)