A105122 -id:A105122 - cp's OEIS frontend

A186689 Numbers n such that n^4 + 1 is a semiprime.

3, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22, 23, 26, 29, 30, 32, 35, 36, 38, 39, 40, 42, 50, 52, 57, 58, 61, 62, 65, 68, 71, 72, 73, 78, 81, 84, 86, 92, 94, 98, 100, 102, 103, 105, 108, 112, 113, 114, 115, 116, 119, 120, 122, 124, 128, 129, 130, 138, 146, 148, 152, 153, 158

Offset: 1

Michel Lagneau, Feb 25 2011

nonn

Corresponding semiprimes n^4+1 are in A186688.

			3 is in the sequence because 3^4 + 1 = 82 = 2*41 is semiprime.

Robert Price, Table of n, a(n) for n = 1..1500

Cf. A001358, A006313, A085722, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282, A186688.

SemiPrimeQ[ n_] := (n > 1) && (2 == Plus @@ (Transpose[FactorInteger[n]][[2]]));
  Select[Range[300], SemiPrimeQ[#^4 + 1] &]
Select[Range[200],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Jan 27 2013 *)

A261460 Numbers k such that k^11-1 is a semiprime.

Original entry on oeis.org

2, 20, 30, 60, 212, 224, 258, 272, 390, 398, 480, 504, 654, 770, 812, 1040, 1194, 1448, 1698, 1748, 1874, 2000, 2238, 2274, 2294, 2438, 2522, 2664, 2714, 2790, 2802, 3020, 3138, 3168, 3300, 3392, 3434, 3794, 4160, 4232, 4518, 4722, 4968, 5334, 5654, 5658

Offset: 1

Vincenzo Librandi, Aug 21 2015

After 2, numbers k such that k-1 and k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 are both prime.

Intersection of A008864 and A162862. - Michel Marcus, Aug 21 2015

			20 is in sequence because 20^11-1 = 204799999999999 = 19*10778947368421, where 19 and 10778947368421 are both prime.

Cf. similar sequences listed in A261435.
Cf. A105122.
Cf. A008864, A162862.

IsSemiprime:=func; [n: n in [2..4000] | IsSemiprime(s) where s is n^11- 1];

Mathematica

Select[Range[6000], PrimeOmega[#^11 - 1] == 2 &]

PARI

isok(n)=bigomega(n^11-1)==2 \\ Anders Hellström, Aug 21 2015

A105934 Positive integers n such that n^22 + 1 is semiprime (A001358).

Original entry on oeis.org

116, 176, 184, 300, 444, 470, 584, 690, 696, 950

Offset: 1

Jonathan Vos Post, Apr 26 2005

We have the polynomial factorization: n^22 + 1 = (n^2 + 1) * (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^2+1 is prime and (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1) is prime.

			116^22 + 1 = 2618639792014920380336685706161496723088736257 = 13457 * 194593133091693570657404005808240820620401,
300^22 + 1 = 3138105960900000000000000000000000000000000000000000001 = 90001 * 34867456593815624270841435095165609271008099910001,
950^22 + 1 = 323533544973709366507562922501564025878906250000000000000000000001 = 902501 * 358485525194663902319845543109164450653136395416736380347501.

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..1000] | IsSemiprime(n^22+1)]; // Vincenzo Librandi, Dec 21 2010

Select[Range[1000], PrimeOmega[#^22 + 1]==2&] (* Vincenzo Librandi, May 24 2014 *)

a(n)^22 + 1 is in A001358. a(n)^2+1 is in A000040 and (a(n)^20 - a(n)^18 + a(n)^16 - a(n)^14 + a(n)^12 - a(n)^10 + a(n)^8 - a(n)^6 + a(n)^4 - a(n)^2 + 1) is in A000040.

cp's OEIS Frontend

A186689 Numbers n such that n^4 + 1 is a semiprime.

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A261460 Numbers k such that k^11-1 is a semiprime.

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A105934 Positive integers n such that n^22 + 1 is semiprime (A001358).

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