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A053027
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Odd primes p with 2 zeros in Fibonacci numbers mod p.
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25
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3, 7, 23, 41, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 241, 263, 281, 283, 307, 347, 367, 383, 401, 409, 443, 449, 463, 467, 487, 503, 523, 547, 563, 569, 587, 601, 607, 641, 643, 647, 683, 727, 743, 769, 787, 823, 827, 863, 881, 883, 887, 907, 929
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OFFSET
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1,1
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COMMENTS
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Also, odd primes that divide Lucas numbers of even index. - T. D. Noe, Jul 25 2003
For p > 2, p is in this sequence if and only if 8 divides of A001175(p), and if and only if 4 divides A001177(p). For a proof of the equivalence between A001176(p) = 2 and 4 dividing A001177(p), see Section 2 of my link below.
This sequence contains all primes congruent to 3, 7 (mod 20). This corresponds to case (2) for k = 3 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
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LINKS
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FORMULA
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A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is even. - T. D. Noe, Jul 25 2003
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CROSSREFS
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Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
| m=1 | m=2 | m=3
-----------------------------+-----------+---------+---------
* and also A053032 (primes dividing Lucas numbers of odd index) U {2}
** also primes dividing no Lucas number
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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