cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A105282 Positive integers n such that n^20 + 1 is semiprime (A001358).

Original entry on oeis.org

2, 4, 46, 154, 266, 472, 748, 1434, 1738, 2058, 2204, 2222, 2428, 2478, 2510, 2866, 3132, 3288, 3576, 3688, 3756, 4142, 4506, 4940, 5164, 6252, 6330, 6786, 7180, 7300, 7338, 7416, 7628, 7806, 9270, 9312, 10044, 10722, 10860, 12126, 12422, 12668, 12998, 13350
Offset: 1

Views

Author

Jonathan Vos Post, Apr 25 2005

Keywords

Comments

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

Examples

			2^20 + 1 = 1048577 = 17 * 61681,
4^20 + 1 = 1099511627777 = 257 * 4278255361,
46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,
1434^20 + 1 =
1352019721694375552250489804528860551814233886722212960509362177 =
4228599998737 * 319732233386510278346888399489424537759394853595121.

Links

Crossrefs

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

Programs

  • Magma
    IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1000] | IsSemiprime(n^20+1)] // Vincenzo Librandi, Dec 21 2010
  • Mathematica
    Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* Robert Price, Mar 09 2015 *)

Formula

a(n)^20 + 1 is semiprime (A001358).

Extensions

a(9)-a(44) from Robert Price, Mar 09 2015

A261436 Numbers k such that k^7-1 is a semiprime.

Original entry on oeis.org

3, 6, 14, 38, 60, 68, 72, 80, 128, 158, 180, 192, 264, 282, 294, 350, 450, 464, 548, 660, 710, 734, 798, 822, 878, 912, 942, 984, 998, 1052, 1188, 1194, 1224, 1280, 1284, 1382, 1424, 1482, 1494, 1512, 1550, 1554, 1572, 1608, 1622, 1668, 1700, 1710, 1790, 1802
Offset: 1

Views

Author

Vincenzo Librandi, Aug 21 2015

Keywords

Comments

Numbers k such that k-1 and k^6+k^5+k^4+k^3+k^2+k+1 are both prime.
Intersection of A008864 and A100330. - Michel Marcus, Aug 21 2015

Examples

			3 is in sequence because 3^7-1 = 2186 = 2*1093, where 2 and 1093 are both prime.

Links

Crossrefs

Cf. similar sequences listed in A261435.
Cf. A105041.
Cf. A008864, A100330.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..5000] | IsSemiprime(s) where s is n^7- 1];
  • Maple
    with(numtheory): A261436:=n->`if`(bigomega(n^7-1)=2, n, NULL): seq(A261436(n), n=1..2000); # Wesley Ivan Hurt, Aug 21 2015
select(n -> isprime(n-1) and isprime(n^6+n^5+n^4+n^3+n^2+n+1), [3,(2*i $i=2..10000)]); # Robert Israel, Aug 21 2015
  • Mathematica
    Select[Range[5000], PrimeOmega[#^7 - 1] == 2 &]
  • PARI
    isok(n)=bigomega(n^7-1)==2 \\ Anders Hellström, Aug 21 2015

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

Original entry on oeis.org

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

Views

Author

Michel Lagneau, Feb 25 2011

Keywords

Comments

Corresponding semiprimes n^4+1 are in A186688.

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

Original entry on oeis.org

61273, 109441, 160213, 274501, 275473, 311593, 360673, 394201, 477181, 486061, 514993, 522085, 617137, 620053, 715477, 725485, 803833, 812677, 847117, 1063585, 1146913, 1182577, 1215865, 1232917, 1409425, 1508113, 1587241, 1768993, 1863073, 1895413, 2085517, 2095177
Offset: 1

Views

Author

K. D. Bajpai, Dec 16 2019

Keywords

Comments

a(2620) = 530079693 is the first multiple of 3 in this sequence; there are no multiples of 2. - Charles R Greathouse IV, Dec 20 2019

Examples

			a(1) = 61273:
  61273 + 6^0  =    61274 =   2 *  30637;
  61273 + 6^1  =    61279 = 233 *    263;
  61273 + 6^2  =    61309 =  37 *   1657;
  61273 + 6^3  =    61489 =  17 *   3617;
  61273 + 6^4  =    62569 =  13 *   4813;
  61273 + 6^5  =    69049 =  29 *   2381;
  61273 + 6^6  =   107929 =  37 *   2917;
  61273 + 6^7  =   341209 =  11 *  31019;
  61273 + 6^8  =  1740889 = 197 *   8837;
  61273 + 6^9  = 10138969 =  89 * 113921;
all ten results are semiprime.

Links

Crossrefs

Subsequence of A076274.
Cf. A001358, A082919, A096173, A104238, A105041.

Programs

  • Magma
    f:=func; [k:k in [1..2100000]|forall{m:m in [0..9]|f(k+6^m)}]; // Marius A. Burtea, Dec 20 2019
  • Mathematica
    fX[n_] = PrimeOmega[n] == 2; Select[Range[2000000], AllTrue[# + 6^{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, fX] &]
  • PARI
    issemi(n)=bigomega(n)==2
is(n)=for(t=0,9, if(!issemi(n+6^t), return(0))); 1 \\ Charles R Greathouse IV, Dec 20 2019

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

Views

Author

Jonathan Vos Post, Apr 26 2005

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

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

Formula

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.

A279272 Numbers k such that k^7 - 1 and k^7 + 1 are semiprimes.

Original entry on oeis.org

72, 282, 9000, 13932, 19212, 22158, 49920, 65538, 72228, 78888, 144408, 169320, 201492, 201828, 218460, 234540, 270030, 296478, 325080, 355008, 365748, 411000, 448872, 461052, 484152, 504618, 555522, 558252, 586362, 622620, 674058, 981810, 1067490, 1095792
Offset: 1

Views

Author

Vincenzo Librandi, Dec 09 2016

Keywords

Comments

Since k^7 - 1 = (k-1)*(k^6 + k^5 + k^4 + k^3 + k^2 + k + 1) and k^7 + 1 = (k+1)*(k^6 - k^5 + k^4 - k^3 + k^2 - k + 1) (and since there is no term less than 3, so k-1 must have at least one prime factor), this sequence lists the numbers k such that k-1, k+1, k^6 + k^5 + k^4 + k^3 + k^2 + k + 1, and k^6 - k^5 + k^4 - k^3 + k^2 - k + 1 are all prime. - Jon E. Schoenfield, Dec 14 2016

Crossrefs

Cf. A105041, A108278, A261436, A268043, A276905.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [4..10000] | IsSemiprime(n^7-1)and IsSemiprime(n^7+1)];
  • Mathematica
    Select[Range[100000], PrimeOmega[#^7 - 1] == PrimeOmega[#^7 + 1]== 2 &]

Extensions

More terms from Jon E. Schoenfield, Dec 14 2016
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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