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A105237 Positive integers n such that n^13 + 1 is semiprime. 13
2, 22, 108, 126, 180, 256, 336, 490, 630, 652, 660, 682, 708, 760, 828, 862, 882, 1030, 1038, 1128, 1162, 1216, 1318, 1450, 1612, 1930, 1950, 2010, 2236, 2268, 2380, 2436, 2658, 2752, 2800, 2962, 2998, 3036, 3048, 3318, 3672, 3922, 4152, 4396, 4506, 4816 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

We have the polynomial factorization: n^13+1 = (n+1) * (n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n+1 is prime and n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 is prime.

LINKS

Robert Price, Table of n, a(n) for n = 1..1036

EXAMPLE

2^13+1 = 8193 = 3 * 2731,

22^13+1 = 282810057883082753 = 23 * 12296089473177511,

1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.

MATHEMATICA

Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)

PROG

(MAGMA) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1600]|IsSemiprime(n^13+1)] // Vincenzo Librandi, Dec 21 2010

CROSSREFS

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

Sequence in context: A212894 A281647 A281140 * A325948 A216801 A083833

Adjacent sequences:  A105234 A105235 A105236 * A105238 A105239 A105240

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Apr 12 2005

EXTENSIONS

a(19)-a(24) from Vincenzo Librandi, Dec 21 2010

STATUS

approved

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Last modified April 11 18:59 EDT 2021. Contains 342888 sequences. (Running on oeis4.)