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 1-7 of 7 results.

A101790 Numbers k such that 4*k-1, 8*k-1 and 16*k-1 are all primes.

Original entry on oeis.org

3, 45, 90, 180, 255, 258, 363, 378, 453, 483, 615, 675, 705, 873, 885, 978, 1350, 1533, 1770, 1788, 2673, 2793, 2868, 3030, 3225, 3240, 4203, 4290, 4548, 4830, 4998, 5103, 5253, 5295, 5568, 5775, 5955, 6060, 6138, 6870, 7383, 7713, 8133, 8370, 8580, 9000
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

Keywords

Examples

			4*3 - 1 = 11, 8*3 - 1 = 23 and 16*3 - 1 = 47 are primes, so 3 is a term.

Links

Crossrefs

Cf. A002515, A101791, A101792, A101793.
Subsequence of A005099 and A005122.
Subsequences: A101794, A101994.

Programs

  • Magma
    [n: n in [0..10000] | IsPrime(4*n-1) and IsPrime(8*n-1) and IsPrime(16*n-1)]; // Vincenzo Librandi, Nov 17 2010
  • Mathematica
    Select[Range[10^4], And @@ PrimeQ[2^Range[2, 4]*# - 1] &] (* Amiram Eldar, May 12 2024 *)
  • PARI
    is(k) = isprime(4*k-1) && isprime(8*k-1) && isprime(16*k-1); \\ Amiram Eldar, May 12 2024

A101997 Primes of the form 16*k-1 such that 4*k-1, 8*k-1, 32*k-1 and 64*k-1 are also primes.

Original entry on oeis.org

719, 214559, 253679, 507359, 508559, 1017119, 1184399, 1363679, 2429279, 3242159, 4276799, 4490639, 6394799, 6486479, 7283999, 7464959, 7650719, 7683839, 8181359, 8553599, 8631599, 8981279, 9112319, 9428879, 10671119
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

Keywords

Examples

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 719 is a term.

Links

Crossrefs

Cf. A101994, A101995, A101996, A101998, A101999.
Subsequence of A127576, A101793 and A101797.

Programs

  • Mathematica
    Select[With[{c=2^Range[2,6]},Table[c n-1,{n,700000}]],AllTrue[#,PrimeQ]&][[All,3]] (* Harvey P. Dale, Nov 29 2018 *)
  • PARI
    is(k) = if(k % 16 == 15, my(m = k\16 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

Formula

a(n) = 16*A101994(n) - 1 = 4*A101995(n) + 3 = 2*A101996(n) + 1. - Amiram Eldar, May 13 2024

A101797 Primes of the form 16*k-1 such that 4*k-1, 8*k-1 and 32*k-1 are also primes.

Original entry on oeis.org

719, 1439, 10799, 14159, 48479, 68639, 109919, 214559, 231359, 253679, 285599, 298799, 329999, 350159, 405599, 429119, 430799, 451679, 488399, 491279, 507359, 508559, 533999, 557759, 666959, 671039, 918959, 1014719, 1017119, 1148879
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

Keywords

Examples

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 719 is a term.

Links

Crossrefs

Cf. A002515, A101794, A101795, A101796, A101798.
Subsequence of A127576 and A101793.
Subsequence: A101997.

Programs

  • Mathematica
    16#-1&/@Select[Range[80000],AllTrue[#*2^Range[2,5]-1,PrimeQ]&] (* Harvey P. Dale, Apr 25 2015 *)
  • PARI
    is(k) = if(k % 16 == 15, my(m = k\16 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

Formula

a(n) = 16*A101794(n) - 1 = 4*A101795(n) + 3 = 2*A101796(n) + 1. - Amiram Eldar, May 13 2024

A101791 Primes of the form 4*k-1 such that 8*k-1 and 16*k-1 are also primes.

Original entry on oeis.org

11, 179, 359, 719, 1019, 1031, 1451, 1511, 1811, 1931, 2459, 2699, 2819, 3491, 3539, 3911, 5399, 6131, 7079, 7151, 10691, 11171, 11471, 12119, 12899, 12959, 16811, 17159, 18191, 19319, 19991, 20411, 21011, 21179, 22271, 23099, 23819
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

Keywords

Examples

			4*3-1 = 11, 8*3-1 = 23 and 16*3-1 = 47 are primes, so 11 is a term.

Links

Crossrefs

Cf. A002515, A101790, A101792, A101793.
Subsequence of A002145.
Subsequences: A101795, A101995.

Programs

  • Mathematica
    p4816Q[n_]:=Module[{nn=(n+1)/4},And@@PrimeQ[{n,8nn-1,16nn-1}]]; Select[ 4*Range[6000]-1,p4816Q] (* Harvey P. Dale, Nov 25 2011 *)
  • PARI
    is(k) = if(k % 4 == 3, my(m = k\4 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1), 0); \\ Amiram Eldar, May 13 2024

Formula

a(n) = 4*A101790(n) - 1. - Amiram Eldar, May 13 2024

A101792 Primes of the form 8*k-1 such that 4*k-1 and 16*k-1 are also primes.

Original entry on oeis.org

23, 359, 719, 1439, 2039, 2063, 2903, 3023, 3623, 3863, 4919, 5399, 5639, 6983, 7079, 7823, 10799, 12263, 14159, 14303, 21383, 22343, 22943, 24239, 25799, 25919, 33623, 34319, 36383, 38639, 39983, 40823, 42023, 42359, 44543, 46199, 47639, 48479, 49103, 54959
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

Keywords

Examples

			4*3 - 1 = 11, 8*3 - 1 = 23 and 16*3 - 1 = 47 are primes, so 23 is a term.

Links

Crossrefs

Cf. A002515, A101790, A101791, A101793.
Subsequence of A007522.
Subsequences: A101796, A101996.

Programs

  • Mathematica
    8 * Select[Range[10^4], And @@ PrimeQ[2^Range[2, 4]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)
  • PARI
    for(k=1,7000,if(isprime(8*k-1)&&isprime(4*k-1)&&isprime(16*k-1),print1(8*k-1,", "))) \\ Hugo Pfoertner, Sep 07 2021

Formula

a(n) = 8*A101790(n) - 1 = 2*A101791(n) + 1. - Amiram Eldar, May 13 2024

A347560 a(n) is the number of solutions to Conv(b,n)=b where Conv(b,n) denotes the limit of b^^t (mod n) as t goes to infinity.

Original entry on oeis.org

2, 2, 3, 2, 5, 4, 5, 4, 5, 3, 8, 3, 7, 8, 7, 4, 9, 3, 10, 8, 8, 5, 14, 6, 9, 4, 12, 6, 15, 9, 7, 10, 9, 10, 14, 4, 9, 10, 18, 7, 19, 5, 13, 14, 10, 3, 20, 10, 13, 12, 14, 7, 9, 12, 20, 10, 13, 7, 28, 9, 15, 21, 11, 17, 24, 10, 14, 13, 22, 15, 24, 7, 9, 17, 17, 20, 24, 10, 28
Offset: 2

Views

Author

Bernat Pagès Vives, Sep 06 2021

Keywords

Comments

Writing n = m^(2k), a(n) >= 2^A001221(n) + m^k - 1.
Writing n = m^(2k+1), a(n) >= 2^A001221(n) + m^k - 1.
If n is in A101793, then a(n) = 3.
It appears that a(n) = 2 only for n = 2, 3, 5.
It appears that a(n) = 3 only for n = 4, 11, 13, 19 and for n in A101793.
It is not known whether there exist infinitely many numbers n satisfying a(n) = 3.

Examples

			For n = 100, pick b = 3.
3^^1 ==  3 (mod 100)
3^^2 == 27 (mod 100)
3^^3 == 87 (mod 100)
3^^4 == 87 (mod 100)
3^^5 == 87 (mod 100)
...
It can be proved that the sequence converges to 87, so Conv(3,100) = 87. Since b = 3 does not satisfy Conv(b,100) = b, this value is not counted in a(100).
For n = 7, pick b = 2.
2^^1 == 2 (mod 7)
2^^2 == 4 (mod 7)
2^^3 == 2 (mod 7)
2^^4 == 2 (mod 7)
2^^5 == 2 (mod 7)
...
It can be proved that the sequence converges to 2, so Conv(2,7) = 2. Thus, 2 is a solution for a(7). The other 3 solutions are 0, 1 and 4 giving a total of a(7) = 4 solutions.

Links

Crossrefs

Cf. A347561, A343073, A000040, A101793, A183613, A001221.

Programs

  • Mathematica
    Conv[b_,n_] :=
Which[
Mod[b,n]==0,Return[0],
Mod[b,n]==1,Return[1],
GCD[b,n]==1,Return[PowerMod[b,Conv[b,MultiplicativeOrder[b,n]],n]],
True,Return[PowerMod[b,EulerPhi[n]+Conv[b,EulerPhi[n]],n]]
]
a[n_] := Count[Table[Conv[b,n]==b,{b,0,n-1}],True]
  • PARI
    conv(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^conv(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + conv(b, eulerphi(n))));}
a(n) = sum(b=0, n-1, conv(b, n) == b); \\ Michel Marcus, Sep 13 2021

A347561 Numbers m such that Conv(b,m) = b has a unique nontrivial solution (b = 0 and b = 1 are considered trivial solutions). Here, Conv(b,m) denotes the limit of b^^t (mod m) as t goes to infinity.

Original entry on oeis.org

4, 11, 13, 19, 47, 719, 1439, 2879, 4079, 4127, 5807, 6047, 7247, 7727, 9839, 10799, 11279, 13967, 14159, 15647, 21599, 24527, 28319, 28607, 42767, 44687, 45887, 48479, 51599, 51839, 67247, 68639, 72767, 77279, 79967, 81647, 84047, 84719, 89087
Offset: 1

Views

Author

Bernat Pagès Vives, Sep 06 2021

Keywords

Comments

A101793 is a subsequence.
It appears that a(n) = A101793(n-4) for n>=5.
Except for n = 1, a(n) is prime.

Examples

			For a(2), we have:
Conv(2,11) = 9
Conv(3,11) = 9
Conv(4,11) = 4
Conv(5,11) = 1
Conv(6,11) = 5
Conv(7,11) = 2
Conv(8,11) = 3
Conv(9,11) = 5
Conv(10,11) = 1
Therefore, the only solution is Conv(4,11) = 4.

Links

Crossrefs

Cf. A347560, A343073, A000040, A101793, A183613.

Programs

  • Mathematica
    Conv[b_,m_] :=
Which[
Mod[b,m]==0,Return[0],
Mod[b,m]==1,Return[1],
GCD[b,m]==1,Return[PowerMod[b,Conv[b,MultiplicativeOrder[b,m]],m]],
True,Return[PowerMod[b,EulerPhi[m]+Conv[b,EulerPhi[m]],m]]
]
a[m_] := Count[Table[Conv[b,m]==b,{b,0,m-1}],True]
Table[If[a[i]==3,i,## &[]],{i,2,800}]
  • PARI
    conv(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^conv(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + conv(b, eulerphi(n))));}
isok(m) = sum(b=2, m-1, conv(b, m) == b) == 1; \\ Michel Marcus, Sep 13 2021
Showing 1-7 of 7 results.

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