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A105142
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Positive integers n such that n^12 + 1 is semiprime.
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7
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2, 6, 34, 46, 142, 174, 204, 238, 312, 466, 550, 616, 690, 730, 1136, 1280, 1302, 1330, 1486, 1586, 1610, 1638, 1644, 1652, 1688, 1706, 1772, 1934, 1952, 1972, 2040, 2102, 2108, 2142, 2192, 2238, 2250, 2376, 2400, 2554, 2612, 2646, 3006, 3094, 3550, 3642
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| We have the polynomial factorization n^12+1 = (n^4+1) * (n^8 - n^4 + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^4+1 is prime and n^8 - n^4 + 1 is prime.
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FORMULA
| a(n)^12+1 is semiprime A001538. a(n)^4+1 is prime and a(n)^8 - a(n)^4 + 1 is prime.
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EXAMPLE
| 2^12+1 = 4097 = 17 * 241,
6^12+1 = 2176782337 = 1297 * 1678321,
34^12+1 = 2386420683693101057 = 1336337 * 1785792568561,
1136^12+1 = 4618915067251126036363854530631172097 = 1665379926017 * 2773490297975392253706241.
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MATHEMATICA
| Select[ Range@3691, PrimeQ[ #^4 + 1] && PrimeQ[(#^12 + 1)/(#^4 + 1)] &] (* Robert G. Wilson v *)
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CROSSREFS
| Cf. A000040, A001538, A103854, A104238, A105041, A105066, A105078, A105122.
Sequence in context: A062970 A088125 A064940 * A192537 A026966 A026976
Adjacent sequences: A105139 A105140 A105141 * A105143 A105144 A105145
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 09 2005
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EXTENSIONS
| a(16)-a(46) from Robert G. Wilson v (rgwv(at)rgwv.com), Feb 10 2006
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