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 11-18 of 18 results.

A141229 Odd numbers k for which A006694((k-1)/2) = 3.

Original entry on oeis.org

27, 43, 109, 125, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1331, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2197, 2203, 2251, 2341, 2347, 2749, 2917, 3163, 3181, 3229, 3259, 3373, 4027, 4339, 4549, 4597, 4651, 4909
Offset: 1

Views

Author

Vladimir Shevelev, Jun 15 2008

Keywords

Comments

Conjecture. The terms of the sequence have only one prime divisor; moreover, p^3 is in the sequence if and only if p is in A001122.

Links

Crossrefs

Cf. A006694, A002326, A001122, A001133.

Programs

  • Mathematica
    r[n_] := EulerPhi[n]/MultiplicativeOrder[2, n]; Select[Range[5000], Total@(r /@ Divisors[#]) - 1 == 3 &] (* Amiram Eldar, Sep 12 2019 *)
  • PARI
    a006694(n)=sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1;
isok(n) = (n % 2) && (a006694((n-1)/2) == 3); \\ Michel Marcus, Feb 08 2016

Extensions

More terms from Michel Marcus, Feb 08 2016

A152307 Primes p such that the multiplicative order of 2 modulo p is (p-1)/7.

Original entry on oeis.org

1163, 1709, 2003, 3109, 3389, 3739, 5237, 5531, 5867, 7309, 9157, 9829, 10627, 10739, 11117, 11243, 11299, 11411, 11467, 13259, 18803, 20147, 20483, 21323, 21757, 27749, 27763, 29947, 30773, 31123, 31627, 32803, 33461, 33587, 34469
Offset: 1

Views

Author

Klaus Brockhaus, Dec 02 2008

Keywords

Links

Crossrefs

Cf. A115591, A001133, A001134, A001135, A001136.

Programs

  • Magma
    [ p: p in PrimesUpTo(34469) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,7) where R is ResidueClassRing(p) ];
  • Mathematica
    Select[Prime[Range[4000]],MultiplicativeOrder[2,#]==(#-1)/7&] (* Harvey P. Dale, Oct 10 2024 *)
  • PARI
    Vec(select(p->((p!=2) && (znorder(Mod(2, p)) == (p-1)/7)), primes(10000))) \\ Michel Marcus, Feb 09 2015

A152308 Primes p such that the multiplicative order of 2 modulo p is (p-1)/8.

Original entry on oeis.org

73, 89, 233, 937, 1217, 1249, 1289, 1433, 1553, 1609, 1721, 1913, 2441, 2969, 3257, 3449, 4049, 4201, 4273, 4297, 4409, 4481, 4993, 5081, 5297, 5689, 6089, 6449, 6481, 6689, 6857, 7121, 7529, 7993, 8081, 8609, 8969, 9137, 9281, 9769, 10337, 10369
Offset: 1

Views

Author

Klaus Brockhaus, Dec 02 2008

Keywords

Links

Crossrefs

Cf. A115591, A001133, A001134, A001135, A001136.

Programs

  • Magma
    [ p: p in PrimesUpTo(10369) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,8) where R is ResidueClassRing(p) ];
  • Mathematica
    okQ[p_] := MultiplicativeOrder[2, p] == (p-1)/8;
Select[Prime[Range[2000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)
  • PARI
    Vec(select(p->((p!=2) && (znorder(Mod(2, p)) == (p-1)/8)), primes(10000))) \\ Michel Marcus, Feb 09 2015

A152309 Primes p such that the multiplicative order of 2 modulo p is (p-1)/9.

Original entry on oeis.org

397, 7867, 10243, 10333, 12853, 13789, 14149, 14293, 14563, 15643, 17659, 18379, 18541, 21277, 21997, 23059, 23203, 26731, 27739, 29179, 29683, 31771, 34147, 35461, 35803, 36541, 37747, 39979, 40213, 40429, 41131, 41491, 44029, 44101
Offset: 1

Views

Author

Klaus Brockhaus, Dec 02 2008

Keywords

Links

Crossrefs

Cf. A115591, A001133, A001134, A001135, A001136.

Programs

  • Magma
    [ p: p in PrimesUpTo(44101) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,9) where R is ResidueClassRing(p) ];
  • Mathematica
    okQ[p_] := MultiplicativeOrder[2, p] == (p-1)/9;
Select[Prime[Range[10000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)
  • PARI
    Vec(select(p->((p!=2) && (znorder(Mod(2, p)) == (p-1)/9)), primes(10000))) \\ Michel Marcus, Feb 09 2015

A152310 Primes p such that the multiplicative order of 2 modulo p is (p-1)/10.

Original entry on oeis.org

151, 241, 431, 641, 911, 3881, 4751, 4871, 5441, 5471, 5641, 5711, 6791, 6871, 8831, 9041, 9431, 10711, 12721, 13751, 14071, 14431, 14591, 15551, 16631, 16871, 17231, 17681, 17791, 18401, 19031, 19471, 21401, 25111, 25391, 25561, 26921, 27031
Offset: 1

Views

Author

Klaus Brockhaus, Dec 02 2008

Keywords

Links

Crossrefs

Cf. A115591, A001133, A001134, A001135, A001136.

Programs

  • Magma
    [ p: p in PrimesUpTo(27031) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,10) where R is ResidueClassRing(p) ];
  • Mathematica
    okQ[p_] := MultiplicativeOrder[2, p] == (p-1)/10;
Select[Prime[Range[10000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)
  • PARI
    Vec(select(p->((p!=2) && (znorder(Mod(2, p)) == (p-1)/10)), primes(10000))) \\ Michel Marcus, Feb 09 2015

A152311 Primes p such that the multiplicative order of 2 modulo p is (p-1)/11.

Original entry on oeis.org

331, 1013, 4643, 12101, 12893, 16061, 17117, 23893, 25763, 25939, 28403, 30493, 32429, 32957, 34739, 36389, 38149, 39139, 42043, 44771, 45541, 46861, 53923, 57773, 59621, 60611, 81533, 85229, 87187, 89123, 92357, 96493, 100981, 105227
Offset: 1

Views

Author

Klaus Brockhaus, Dec 02 2008

Keywords

Links

Crossrefs

Cf. A115591, A001133, A001134, A001135, A001136.

Programs

  • Magma
    [ p: p in PrimesUpTo(105227) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,11) where R is ResidueClassRing(p) ];
  • Mathematica
    okQ[p_] := MultiplicativeOrder[2, p] == (p-1)/11;
Select[Prime[Range[20000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)
  • PARI
    Vec(select(p->((p!=2) && (znorder(Mod(2, p)) == (p-1)/11)), primes(20000))) \\ Michel Marcus, Feb 09 2015

A246755 Numbers of the form 2k - 1 such that A246702(k) = 3.

Original entry on oeis.org

15, 33, 43, 45, 69, 75, 87, 99, 109, 135, 141, 157, 159, 177, 207, 213, 225, 229, 249, 261, 277, 283, 297, 303, 307, 321, 363, 375, 393, 405, 423, 447, 477, 499, 501, 519, 531, 537, 573, 591, 621, 639, 643, 675, 681, 691, 717, 733, 739, 747, 783, 789, 807, 811
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 15 2014

Keywords

Comments

Composites in this sequence: 15, 33, 45, 69, 75, 87, 99, 135, 141, 159, 177, 207, 213, 225, 249, 261, 297, 303, 321, 363, 375, 393, 405, 423, 447, 477, ...

Examples

			A246702(8) = 3 for the first time, hence a(1) = 2*8 - 1 = 15.

Crossrefs

Cf. Numbers of the form 2k - 1 such that A246702(k) = m: number 1 (m = 0), A167791 (m = 1), A246717 (m = 2), this sequence (m = 3), A001133 (primes in this sequence).

Programs

  • PARI
    is(k) = (m=Mod(k%2, k*k)) && sum(i=1, k*k-1, m*=2; m==1) == 3; \\ Jinyuan Wang, May 15 2020

Extensions

More terms from and terms corrected by Jinyuan Wang, May 15 2020

A384184 Order of the permutation of {0,...,n-1} formed by successively swapping elements at i and 2*i mod n, for i = 0,...,n-1.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 2, 8, 3, 4, 5, 4, 6, 4, 6, 16, 4, 6, 9, 8, 4, 10, 28, 8, 10, 12, 9, 8, 14, 12, 12, 32, 5, 8, 70, 12, 18, 18, 24, 16, 10, 8, 7, 20, 210, 56, 126, 16, 110, 20, 60, 24, 26, 18, 120, 16, 9, 28, 29, 24, 30, 24, 60, 64, 6, 10, 33, 16
Offset: 1

Views

Author

Mia Boudreau, May 29 2025

Keywords

Comments

a(2*n) = 2*a(n) since the cycle lengths of the permutation with size 2*n is effectively that of size n twice, doubled. Thus, the LCM/order is doubled.

Examples

			For n = 11, the permutation is {0,3,4,7,8,1,2,9,10,5,6} and it has order a(11) = 5.

Links

Crossrefs

Cf. A003418, A000027, A051732, A004626, A065119, A001122, A155072, A001133, A001134, A001135, A225759.

Programs

  • Python
    from sympy.combinatorics import Permutation
def a(n):
   L = list(range(n))
   for i in range(n):
       if (j:= (i << 1) % n) != i:
           L[i],L[j] = L[j],L[i]
   return Permutation(L).order() # Darío Clavijo, Jun 05 2025

Formula

a(2*n) = 2*a(n).
a(2^n) = 2^n.
Conjecture: a(2^n + 2^x) = 2^n * (x-n) if x > n.
a(2^n - 1) = A003418(n-1).
s(2^n + 1) = A000027(n).
a(2*n - 1) = A051732(n).
a(A004626(n)) % 2 = 1.
a(A065119(n)) = n/3.
a(A001122(n)) = (n-1) / 2.
a(A155072(n)) = (n-1) / 4.
a(A001133(n)) = (n-1) / 6.
a(A001134(n)) = (n-1) / 8.
a(A001135(n)) = (n-1) / 10.
a(A225759(n)) = (n-1) / 16.
Previous Showing 11-18 of 18 results.

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