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A103854 Positive integers n such that n^6 + 1 is semiprime. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Robert Price, Table of n, a(n) for n = 1..1134

FORMULA

a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Sequence in context: A156800 A210772 A125859 * A126941 A188495 A038077

Adjacent sequences:  A103851 A103852 A103853 * A103855 A103856 A103857

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Mar 31 2005

EXTENSIONS

More terms from Robert G. Wilson v, May 26 2006

STATUS

approved

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Last modified April 14 12:11 EDT 2021. Contains 342949 sequences. (Running on oeis4.)