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A103854
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Positive integers n such that n^6 + 1 is semiprime.
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15
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2, 4, 10, 36, 56, 94, 126, 224, 260, 270, 300, 350, 686, 716, 780, 1036, 1070, 1080, 1156, 1174, 1210, 1394, 1416, 1434, 1440, 1460, 1524, 1550, 1576, 1616, 1654, 1660, 1700, 1756, 1860, 1980, 2054, 2084, 2096, 2116, 2224, 2454, 2600, 2664, 2770, 2864
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OFFSET
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1,1
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COMMENTS
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n^6+1 can only be prime when n = 1, n^6+1 = 2. This is because the sum of cubes formula gives the polynomial factorization n^6+1 = (n^2+1) * (n^4 - n^2 + 1). Hence n^6+1 can only be semiprime when both (n^2+1) and (n^4 - n^2 + 1) are primes.
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LINKS
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FORMULA
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a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.
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EXAMPLE
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n n^6+1 = (n^2+1) * (n^4 - n^2 + 1)
2 65 = 5 * 13
4 4097 = 17 * 241
10 1000001 = 101 * 9901
36 2176782337 = 1297 * 1678321
56 30840979457 = 3137 * 9831361
94 689869781057 = 8837 * 78066061
126 4001504141377 = 15877 * 252031501
224 126324651851777 = 50177 * 2517580801
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MATHEMATICA
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semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ 2Range@1526, semiprimeQ[ #^6 + 1] &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[200000], PrimeQ[#^2 + 1] && PrimeQ[(#^6 + 1)/(#^2 + 1)] &] (* Robert Price, Mar 11 2015 *)
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PROG
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CROSSREFS
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Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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