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.

Showing 1-10 of 13 results. Next

A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

Original entry on oeis.org

7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993
Offset: 1

Views

Author

Labos Elemer, Dec 20 2000

Keywords

Comments

Prime numbers n such that cos(2*Pi/n) is an algebraic number of a 3-smooth degree, but not a 2-smooth degree. - Artur Jasinski, Dec 13 2006
From Antonio M. Oller-Marcén, Sep 24 2009: (Start)
In this case gcd(a,b) is a power of 2.
A regular polygon of n sides is constructible by paper folding if and only if n=2^r3^sp_1...p_t with p_i being distinct primes of this kind. (End)
Primes in A005109 but not in A092506. - R. J. Mathar, Sep 28 2012
Conjecture: these are the only solutions >=7 to the equation A000010(x) + A000010(x-1) = floor((4*x-3)/3). - Benoit Cloitre, Mar 02 2018
These are also called Pierpont primes. - Harvey P. Dale, Apr 13 2019

Links

Crossrefs

Cf. A033845, A000423, A125866, A217035.

Programs

  • Maple
    N:= 10^10: # to get all terms <= N+1
sort(select(isprime, [seq(seq(1+2^a*3^b, a=1..ilog2(N/3^b)), b=1..floor(log[3](N)))])); # Robert Israel, Mar 02 2018
  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {3} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
mx = 1500000; s = Sort@ Flatten@ Table[1 + 2^j*3^k, {j, Log[2, mx]}, {k, Log[3, mx/2^j]}]; Select[s, PrimeQ] (* Robert G. Wilson v, Sep 28 2012 *)
Select[Prime[Range[114000]],FactorInteger[#-1][[All,1]]=={2,3}&] (* Harvey P. Dale, Apr 13 2019 *)

Formula

Primes of the form 1 + A033845(n).

A125878 Duplicate of A066674.

Original entry on oeis.org

3, 7, 11, 29, 23, 53, 103, 191, 47, 59, 311, 149, 83, 173, 283, 107, 709, 367, 269, 569, 293, 317, 167, 179, 389, 607, 619, 643, 1091, 227, 509, 263, 823, 557, 1193, 907, 1571, 653, 2339, 347, 359, 1087, 383, 773, 3547, 797, 2111, 2677, 5449, 2749, 467
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

Original name was: a(n) = the least number k such that cos(2pi/k) is an algebraic number of a prime(n)-smooth degree, but not prime(n-1)-smooth.
Comments from N. J. A. Sloane, Jan 07 2013: (Start)
This is a duplicate of A066674. This follows from the following argument. The degree of the minimal polynomial of cos(2*Pi/k) is phi(k)/2, where phi is Euler's totient function. Then a(n) is the least number k such that prime(n) is the largest prime dividing phi(k) and prime(n-1) does not divide phi(k)/2. For the rest of the proof see Bjorn Poonen's remarks in A066674.
It also seems likely that this is the same as A035095, but this is an open problem.
Conjecture: this sequence contains only primes (this would follow if this is indeed the same as A035095).
(End)

References

Crossrefs

Cf. A066674, A035095, A125866-A125877.

Extensions

Edited by Don Reble, Apr 24 2007
Minor edits by Ray Chandler, Oct 20 2011

A125867 Numbers k such that p=6k+1 is prime and cos(2*Pi/p) is an algebraic number of a 3-smooth degree, but not 2-smooth.

Original entry on oeis.org

1, 2, 3, 6, 12, 16, 18, 27, 32, 72, 81, 96, 128, 192, 216, 243, 432, 486, 576, 648, 1728, 2048, 2916, 3072, 6561, 8748, 23328, 24576, 34992, 55296, 78732, 104976, 124416, 131072, 139968, 165888, 196608, 248832, 294912, 331776, 442368, 839808
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

Numbers k such that p=6k+1 is prime and the greatest prime divisor of p-1 is 3.

Crossrefs

Cf. A024899, A058383, A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[6n+1]][[ -1]], 1]=={3} && PrimeQ[6n+1],Print[n]],{n,1,100000}]

Extensions

Edited by Don Reble, Apr 24 2007

A125875 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 13-smooth degree, but not 11-smooth.

Original entry on oeis.org

53, 79, 131, 157, 159, 169, 237, 265, 313, 371, 393, 395, 471, 477, 507, 521, 547, 553, 583, 655, 677, 689, 711, 785, 795, 845, 859, 869, 901, 911, 917, 937, 939, 1007, 1027, 1093, 1099, 1113, 1171, 1179, 1183, 1185, 1219, 1249, 1301, 1325, 1343
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

A regular polygon of a(n) sides can be constructed if one also has an angle trisector, 5-, 7-, 11- and 13-sector.

Crossrefs

Cf. A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[2n+1]][[ -1]],1]=={13},Print[2n+1]],{n,1,10000}]

Extensions

Edited by Don Reble, Apr 24 2007

A125877 Numbers k such that p=26*k+1 is prime and cos(2*Pi/p) is an algebraic number of a 13-smooth degree, but not 11-smooth.

Original entry on oeis.org

2, 3, 5, 6, 12, 20, 21, 26, 33, 35, 36, 42, 45, 48, 50, 72, 75, 77, 78, 80, 90, 98, 105, 110, 120, 125, 128, 132, 135, 143, 147, 156, 182, 192, 225, 231, 252, 260, 275, 288, 297, 308, 315, 330, 336, 351, 363, 378, 390, 392, 405, 441, 450, 455, 486, 500, 507, 512
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

Numbers k such that p=26*k+1 is prime and the greatest prime divisor of p-1 is 13.

Links

Crossrefs

Cf. A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[26n+1]][[ -1]],1]=={13} && PrimeQ[26n+1],Print[n]],{n,1,10000}]
(* or *)
Select[Range[600],PrimeQ[26#+1]&&FactorInteger[26#][[-1,1]]==13&] (* Harvey P. Dale, Jun 01 2019 *)

Extensions

Edited by Don Reble, Apr 24 2007

A125874 Numbers k such that p=22*k+1 is prime and cos(2*Pi/p) is an algebraic number of an 11-smooth degree, but not 7-smooth.

Original entry on oeis.org

1, 3, 4, 9, 15, 16, 18, 21, 28, 30, 33, 40, 45, 60, 64, 66, 81, 96, 99, 105, 108, 121, 135, 144, 150, 154, 165, 168, 175, 189, 198, 210, 225, 240, 243, 250, 288, 294, 324, 336, 343, 378, 396, 420, 448, 450, 490, 495, 525, 528, 550, 616, 625, 640, 675, 700, 726
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

Numbers k such that p=22*k+1 is prime and the greatest prime divisor of p-1 is 11.

Crossrefs

Cf. A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[22n+1]][[ -1]],1]=={11} && PrimeQ[22n+1],Print[n]],{n,1,10000}]

Extensions

Edited by Don Reble, Apr 24 2007

A125868 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 5-smooth degree, but not 3-smooth.

Original entry on oeis.org

11, 25, 31, 33, 41, 55, 61, 75, 77, 93, 99, 101, 123, 125, 143, 151, 155, 165, 175, 181, 183, 187, 205, 209, 217, 225, 231, 241, 251, 271, 275, 279, 287, 297, 303, 305, 325, 341, 369, 375, 385, 401, 403, 407, 425, 427, 429, 451, 453, 465, 475, 495, 505, 525
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

A regular polygon of a(n) sides can be constructed if one also has an angle trisector and 5-sector.

Crossrefs

Cf. A058383, A061599, A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[2n+1]][[ -1]],1]=={5},Print[2n+1]],{n,1,1000}]

Extensions

Edited by Don Reble, Apr 24 2007

A125869 Numbers k such that p=10*k+1 is prime and cos(2*Pi/p) is an algebraic number of a 5-smooth degree, but not 3-smooth.

Original entry on oeis.org

1, 3, 4, 6, 10, 15, 18, 24, 25, 27, 40, 54, 60, 64, 75, 81, 120, 160, 162, 180, 216, 225, 300, 400, 405, 480, 486, 648, 768, 810, 864, 900, 960, 972, 1125, 1440, 1536, 1600, 1944, 2160, 2187, 2250, 2304, 2400, 2560, 3240, 3375, 3645, 3750, 4096, 4320
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

Numbers k such that p=10*k+1 is prime and the greatest prime divisor of p-1 is 5.

Crossrefs

Cf. A024912, A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[10n+1]][[ -1]],1]=={5} && PrimeQ[10n+1],Print[n]],{n,1,10000}]

Extensions

Edited by Don Reble, Apr 24 2007

A125870 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 7-smooth degree, but not 5-smooth.

Original entry on oeis.org

29, 43, 49, 71, 87, 113, 127, 129, 145, 147, 197, 203, 211, 213, 215, 245, 261, 281, 301, 319, 337, 339, 343, 355, 377, 379, 381, 387, 421, 435, 441, 449, 473, 491, 493, 497, 539, 551, 559, 565, 591, 609, 631, 633, 635, 637, 639, 645, 673, 701, 725, 731
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

A regular polygon of a(n) sides can be constructed if one also has an angle trisector, 5-sector and 7-sector.

Crossrefs

Cf. A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[2n+1]][[ -1]],1]=={7},Print[2n+1]],{n,1,1000}]

Extensions

Edited by Don Reble, Apr 24 2007

A125871 Numbers k such that p=14*k+1 is prime and cos(2*Pi/p) is an algebraic number of a 7-smooth degree, but not 5-smooth.

Original entry on oeis.org

2, 3, 5, 8, 9, 14, 15, 20, 24, 27, 30, 32, 35, 45, 48, 50, 54, 63, 72, 75, 98, 105, 144, 162, 180, 189, 192, 200, 224, 240, 252, 300, 320, 420, 450, 500, 504, 525, 540, 560, 588, 630, 750, 768, 864, 875, 900, 960, 980, 1029, 1080, 1134, 1215, 1280, 1323
Offset: 1

Views

Author

Artur Jasinski, Dec 13 2006

Keywords

Comments

Numbers k such that p=14*k+1 is prime and the greatest prime divisor of p-1 is 7.

Crossrefs

Cf. A125866-A125878.

Programs

  • Mathematica
    Do[If[Take[FactorInteger[EulerPhi[14n+1]][[ -1]],1]=={7} && PrimeQ[14n+1],Print[n]],{n,1,10000}]

Extensions

Edited by Don Reble, Apr 24 2007
Showing 1-10 of 13 results. Next

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