A104238 -id:A104238 - cp's OEIS frontend

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A104657 Positive integers n such that n^19 + 1 is semiprime (A001358).

2, 10, 28, 106, 190, 292, 556, 756, 858, 906, 1012, 1030, 1032, 1060, 1372, 1450, 1488, 1720, 1722, 1758, 1782, 1822, 1972, 2356, 2436, 2446, 2620, 2748, 2788, 2998, 3186, 3300, 3318, 3360, 3466, 3510, 3822, 3852, 4138, 4326, 4506, 4908, 5236, 5518, 5782

Offset: 1

Jonathan Vos Post, Apr 21 2005

We have the polynomial factorization: n^19 + 1 = (n + 1) * (n^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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.

			2^19 + 1 = 524289 = 3 * 174763,
10^19 + 1 = 10000000000000000001 = 11 * 909090909090909091,
1012^19 + 1 = 125438178100868833265294241234853844232270960601988910249 = 1013 * 1238284087866424810121364671617510801898035149081825373.

Robert Price, Table of n, a(n) for n = 1..1000

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

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1100]|IsSemiprime(n^19+1)]; // Vincenzo Librandi, Mar 10 2015

Select[Range[1000000], PrimeQ[# + 1] && PrimeQ[(#^19 + 1)/(# + 1)] &] (* Robert Price, Mar 10 2015 *)
Select[Range[5800],PrimeOmega[#^19+1]==2&] (* Harvey P. Dale, Feb 15 2019 *)

a(n)^19 + 1 is semiprime (A001358).

a(12)-a(45) from Robert Price, Mar 09 2015

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

Jonathan Vos Post, Apr 25 2005

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.

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

Robert Price, Table of n, a(n) for n = 1..1405

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

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

Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* Robert Price, Mar 09 2015 *)

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

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

A261435 Numbers k such that k^5-1 is a semiprime.

Original entry on oeis.org

12, 24, 30, 44, 62, 68, 74, 110, 164, 198, 308, 492, 572, 594, 662, 728, 770, 824, 854, 860, 864, 942, 954, 968, 1152, 1154, 1284, 1382, 1452, 1668, 1694, 1748, 1760, 1788, 1914, 2090, 2252, 2274, 2340, 2382, 2438, 2448, 2648, 2658, 2664, 2690, 2714, 2790

Offset: 1

Vincenzo Librandi, Aug 20 2015

Numbers k such that k-1 and k^4+k^3+k^2+k+1 are both prime.

			a(1) = 12 because 12^5-1 = 248831 = 11*22621.

Cf. numbers k such that k^p-1 is a semiprime, where p is prime: A096175(p=3), this sequence (p=5), A261436 (p=7), A261460 (p=11).

Cf. A104238.

IsSemiprime:=func; [n: n in [2..5000] | IsSemiprime(s) where s is n^5- 1];

Mathematica

Select[Range[5000], PrimeOmega[#^5 - 1] == 2 &]

PARI

isok(n)=bigomega(n^5-1)==2 \\ Anders Hellström, Aug 20 2015

A108868 Numbers n such that n^5 + 3 is semiprime.

Original entry on oeis.org

1, 2, 4, 6, 11, 14, 18, 19, 24, 31, 32, 38, 40, 46, 50, 55, 59, 70, 74, 76, 84, 92, 96, 100, 115, 119, 128, 139, 148, 150, 151, 154, 155, 158, 164, 178, 184, 200, 203, 204, 206, 210, 230, 236, 238, 239, 242, 248, 256, 263, 272, 278, 284, 295, 299, 304, 306, 310

Offset: 1

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Author

Jonathan Vos Post, Jul 12 2005

Keywords

easy
nonn

Comments

Note that n^5 + 3 is irreducible over integers, unlike n^5 + 1 as in A104238.

Examples

1^5 + 3 = 4 = 2 * 2 2^5 + 3 = 35 = 5 * 7 4^5 + 3 = 1027 = 13 * 79 6^5 + 3 = 7779 = 3 * 2593 11^5 + 3 = 161054 = 2 * 80527 14^5 + 3 = 89 * 6043 100^5 + 3 = 10000000003 = 7 * 1428571429 1000^5 + 3 = 1000000000000003 = 14902357 * 67103479 1000000^5 + 3 = 1000000000000000000000000000003 = 1859827 * 537684419034673655130289.

Crossrefs

Cf. A104238, A108814.

Programs

Maple

with(numtheory): a:=proc(n) if bigomega(n^5+3)=2 then n else fi end: seq(a(n),n=1..400); # Emeric Deutsch, Jul 16 2005

Mathematica

Select[Range[400],PrimeOmega[#^5+3]==2&] (* Harvey P. Dale, Jul 16 2017 *)

Extensions

More terms from Emeric Deutsch, Jul 16 2005

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

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Author

Michel Lagneau, Feb 25 2011

Keywords

nonn

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

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

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 *)

A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

Original entry on oeis.org

12, 1452, 11352, 79398, 146520, 281622, 352110, 536778, 643302, 680988, 723492, 739200, 878988, 992112, 1115268, 1189650, 1397022, 1698378, 1698510, 1728540, 1806222, 2486220, 2873178, 3031578, 3571458, 3946140, 4467012, 4983858, 5064510, 5135658, 5567562, 5753352

Offset: 1

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Author

Gary E. Davis, Sep 21 2016

Keywords

nonn

Crossrefs

Cf. A001358, A108278, A124936, A268043.

Intersection of A104238 and A261435.

Programs

Mathematica

upper=600000; Select[Range[upper], PrimeOmega[#^5 - 1] == PrimeOmega[#^5 + 1] == 2 &]

PARI

isok(n) = (bigomega(n^5-1)==2) && (bigomega(n^5+1)==2); \\ Michel Marcus, Sep 22 2016

Extensions

More terms from Altug Alkan, Sep 30 2016

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

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Author

K. D. Bajpai, Dec 16 2019

Keywords

nonn

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

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Author

Jonathan Vos Post, Apr 26 2005

Keywords

easy
nonn

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.

Previous Showing 11-18 of 18 results.

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A104657 Positive integers n such that n^19 + 1 is semiprime (A001358).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A261435 Numbers k such that k^5-1 is a semiprime.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A108868 Numbers n such that n^5 + 3 is semiprime.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI