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A207262
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a(n) = 2^(4n - 2) + 1.
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8
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5, 65, 1025, 16385, 262145, 4194305, 67108865, 1073741825, 17179869185, 274877906945, 4398046511105, 70368744177665, 1125899906842625, 18014398509481985, 288230376151711745, 4611686018427387905, 73786976294838206465, 1180591620717411303425, 18889465931478580854785, 302231454903657293676545
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OFFSET
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1,1
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COMMENTS
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With the exception of the first term, all these numbers are composite, and in fact are all multiples of 5. The other factors can be considerably larger than 5, as is the case with say, 2^158 + 1. These numbers can be factored as (2^(2n - 1) + 2^n + 1)(2^(2n - 1) - 2^n + 1). For example, 2^6 + 1 = 65 = (2^3 + 2^2 + 1)(2^3 - 2^2 + 1) = 13 * 5.
This formula was discovered by Leon-Francois-Antoine Aurifeuille in 1873. Wells (2005) remarks that knowledge of this formula would have saved Fortune Landry years of work he spent factoring 2^58 + 1.
Aurifeuille actually rediscovered a very special case of the identity 4x^4+1 = (2x^2-2x+1)(2x^2+2x+1), which Euler communicated to Goldbach in 1742. (The Fuss reference is in my book Seminumerical Algorithms, 3rd ed., p. 392; I had cited Aurifeuille in the 1st and 2nd editions.) - Don Knuth, Feb 09 2013
An Engel expansion of 4 to the base 16 as defined in A181565, with the associated series expansion 4 = 16/5 + 16^2/(5*65) + 16^3/(5*65*1025) + 16^4/(5*65*1025*16385) + .... Cf. A087289 and A199561. - Peter Bala, Oct 29 2013
Conjecture: Let m = 4n - 2. a(n) equals the sum of the m-th powers of the divisors of m divided by the sum of the m-th powers of the odd divisors of m. - Ivan N. Ianakiev, Jan 29 2020
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REFERENCES
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David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005) p. 15
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LINKS
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FORMULA
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a(n) = 4^(2n - 1) + 1.
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MATHEMATICA
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2^(4*Range[20] - 2) + 1
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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