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#10 by Charles R Greathouse IV at Thu Sep 08 08:45:17 EDT 2022
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| PROG
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(MAGMAMagma)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
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Discussion
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| Thu Sep 08
| 08:45
| OEIS Server: https://oeis.org/edit/global/2944
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#9 by N. J. A. Sloane at Sat May 24 01:49:26 EDT 2014
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#8 by Vincenzo Librandi at Sat May 24 01:41:54 EDT 2014
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#7 by Vincenzo Librandi at Sat May 24 01:41:43 EDT 2014
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| PROG
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(MAGMA)IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..1000] | IsSemiprime(n^22+1)][From)]; // _Vincenzo V. Librandi, _, Dec 21 2010]
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#6 by Vincenzo Librandi at Sat May 24 01:40:48 EDT 2014
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| MATHEMATICA
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Select[Range[1000], PrimeOmega[#^22 + 1]==2&] (* Vincenzo Librandi, May 24 2014 *)
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| STATUS
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approved
editing
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#5 by Russ Cox at Fri Mar 30 18:40:28 EDT 2012
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| AUTHOR
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_Jonathan Vos Post (jvospost3(AT)gmail.com), _, Apr 26 2005
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Discussion
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| Fri Mar 30
| 18:40
| OEIS Server: https://oeis.org/edit/global/228
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#4 by T. D. Noe at Tue Dec 21 12:22:25 EST 2010
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#3 by Vincenzo Librandi at Tue Dec 21 05:53:51 EST 2010
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| PROG
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(MAGMA)IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..1000] | IsSemiprime(n^22+1)][From V. Librandi, Dec 21 2010]
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| STATUS
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approved
proposed
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#2 by N. J. A. Sloane at Fri Jan 09 03:00:00 EST 2009
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| KEYWORD
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easy,nonn,new
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| AUTHOR
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Jonathan Vos Post (jvospost2jvospost3(AT)yahoogmail.com), Apr 26 2005
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#1 by N. J. A. Sloane at Tue Jul 19 03:00:00 EDT 2005
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| NAME
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Positive integers n such that n^22 + 1 is semiprime (A001358).
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| DATA
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116, 176, 184, 300, 444, 470, 584, 690, 696, 950
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| OFFSET
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1,1
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| COMMENTS
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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.
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| FORMULA
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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.
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| EXAMPLE
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116^22 + 1 = 2618639792014920380336685706161496723088736257 = 13457 * 194593133091693570657404005808240820620401,
300^22 + 1 = 3138105960900000000000000000000000000000000000000000001 = 90001 * 34867456593815624270841435095165609271008099910001,
950^22 + 1 = 323533544973709366507562922501564025878906250000000000000000000001 = 902501 * 358485525194663902319845543109164450653136395416736380347501.
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| CROSSREFS
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Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282.
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| KEYWORD
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easy,nonn
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| AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Apr 26 2005
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| STATUS
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approved
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