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Search: id:A105041
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| A105041 |
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Positive integers n such that n^7 + 1 is semiprime. |
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+0
12
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| 2, 10, 16, 18, 46, 52, 66, 72, 78, 106, 136, 148, 226, 228, 240, 262, 282, 330, 442, 508, 616, 630, 732, 750, 756, 768, 810, 828, 910, 936, 982, 1032
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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We have the polynomial factorization n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1). Hence after the initial n=1 prime, the binomial can at best be semiprime and that only when both (n+1) and (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) are primes.
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FORMULA
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a(n)^7 + 1 is semiprime. a(n)+1 is prime and a(n)^6 - a(n)^5 + a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.
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EXAMPLE
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n n^7+1 = ((n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
2 129 = 3 x 43
10 10000001 = 11 * 909091
16 268435457 = 17 * 15790321
18 612220033 = 19 * 32222107
46 435817657217 = 47 * 9272716111
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CROSSREFS
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Cf. A000040, A001538, A103854, A104238.
Sequence in context: A047187 A048043 A043429 this_sequence A138632 A060658 A054028
Adjacent sequences: A105038 A105039 A105040 this_sequence A105042 A105043 A105044
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 03 2005
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