Peter R. J. Asveld - cp's OEIS frontend

A163780 Terms in A054639 equal to 3 mod 4.

3, 11, 23, 35, 39, 51, 83, 95, 99, 119, 131, 135, 155, 179, 183, 191, 231, 239, 243, 251, 299, 303, 323, 359, 371, 375, 411, 419, 431, 443, 483, 491, 495, 515, 519, 531, 543, 575, 611, 615, 639, 651, 659, 683, 719, 723, 743, 755, 771, 779, 783, 791, 803, 831, 879

Offset: 1

Peter R. J. Asveld, Aug 12 2009

nonn

Previous name was: a(n) is the n-th A^-_1-prime (Archimedes^-_1 prime).

N is A^-_1-prime iff N=3 (mod 4), p=2N+1 is a prime number and -2 generates Z_p^* (the multiplicative group of Z_p) but +2 does not.

P. R. J. Asveld Table of n, a(n) for n=1..3378.
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.

The A^-_1-primes are the T- or Twist-primes congruent 3 (mod 4), these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A_1- and A^+_1-primes, see A163777, A163778 and A163779. Considered as sets the union of A163779 and A163780 equals A163778, the union of A163780 and A163777 is equal to A163781 (dual J_2-primes).

ok(n) = n%4==3 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == n;
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(33)-a(55) from Andrew Howroyd, Nov 11 2017

New name from Andrey Zabolotskiy, Mar 23 2018

A163778 Odd terms in A054639.

Original entry on oeis.org

3, 5, 9, 11, 23, 29, 33, 35, 39, 41, 51, 53, 65, 69, 81, 83, 89, 95, 99, 105, 113, 119, 131, 135, 155, 173, 179, 183, 189, 191, 209, 221, 231, 233, 239, 243, 245, 251, 261, 273, 281, 293, 299, 303, 309, 323, 329, 359, 371, 375, 393, 411, 413, 419, 429

Offset: 1

Peter R. J. Asveld, Aug 11 2009

nonn

Previous name was: The A_1-primes (Archimedes_1 primes).
We have: (1) N is an A_1-prime iff N is odd, p=2N+1 is a prime number and only one of +2 and -2 generates Z_p^* (the multiplicative group of Z_p); (2) N is an A_1-prime iff p=2N+1 is a prime number and exactly one of the following holds: (a) N == 1 (mod 4) and +2 generates Z_p^* but -2 does not, (b) N == 3 (mod 4) and -2 generates Z_p^* but +2 does not.
The A_1-primes are the odd T- or Twist-primes (the T-primes are the same as the Queneau-numbers, A054639). For the related A_0-, A^+_1- and A^-_1-primes, see A163777, A163779 and A163780. Considered as a set, the present sequence is the union of the A^+_1-primes (A163779) and the A^-_1-primes (A163780). It is also equal to the difference of A054639 and the A_0-primes (A163777).

P. R. J. Asveld, Table of n, a(n) for n=1..6706.
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.

Cf. A054639, A163777, A163779, A163780, A294434, A294673.

follow[s_, f_] := Module[{t, k}, t = f[s]; k = 1; While[t>s, k++; t = f[t]]; If[s == t, k, 0]];
okQ[n_] := n>1 && n == follow[1, Function[j, Ceiling[n/2] + (-1)^j*Ceiling[ (j-1)/2]]];
A163778 = Select[Range[1000], okQ] (* Jean-François Alcover, Jun 07 2018, after Andrew Howroyd *)

Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
ok(n)={n>1 && n==Follow(1, j->ceil(n/2) + (-1)^j*ceil((j-1)/2))}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

ok(n)={n>1 && n%2==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%4==3, n, 2*n)}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(33)-a(55) from Andrew Howroyd, Nov 11 2017

New name from Joerg Arndt, Mar 23 2018

A163779 Numbers k of the form 4j + 1 such that 2k + 1 is a prime with primitive root 2.

Original entry on oeis.org

1, 5, 9, 29, 33, 41, 53, 65, 69, 81, 89, 105, 113, 173, 189, 209, 221, 233, 245, 261, 273, 281, 293, 309, 329, 393, 413, 429, 441, 453, 473, 509, 545, 561, 585, 593, 629, 641, 645, 653, 713, 725, 741, 749, 761, 765, 785, 809, 833, 873, 893, 933, 953, 965, 989, 993

Offset: 1

Peter R. J. Asveld, Aug 12 2009

nonn

Previous name was: a(n) is the n-th A^+_1-prime (Archimedes^+_1 prime).

N is A^+_1-prime iff N=1 (mod 4), p=2N+1 is a prime number and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not.

Joerg Arndt, Table of n, a(n) for n = 1..10000 (first 3328 from P. R. J. Asveld)
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.
P. Michael Hutchins, Reworded Definition

The A^+_1-primes are the T- or Twist-primes congruent 1 (mod 4), these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A_1- and A^-_1-primes, see A163777, A163778 and A163780. Considered as sets the union of A163779 and A163780 equals A163778, the union of A163779 and A163777 is equal to A163782 (J_2-primes).

okQ[n_] := Mod[n, 4] == 1 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* Jean-François Alcover, Jun 30 2018, after Andrew Howroyd *)

ok(n) = n%4==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1))==2*n;
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

2 * a(n) + 1 = A213051(n+1). - Joerg Arndt, Mar 23 2018

a(32)-a(55) from Andrew Howroyd, Nov 11 2017

Term 1 prepended and new name from Joerg Arndt, Mar 23 2018

A163777 Even terms in the sequence of Queneau numbers A054639.

Original entry on oeis.org

2, 6, 14, 18, 26, 30, 50, 74, 86, 90, 98, 134, 146, 158, 174, 186, 194, 210, 230, 254, 270, 278, 306, 326, 330, 338, 350, 354, 378, 386, 398, 410, 414, 426, 438, 470, 530, 554, 558, 606, 614, 618, 638, 650, 686, 690, 726, 746, 774, 810, 818, 834, 846, 866, 870

Offset: 1

Peter R. J. Asveld, Aug 11 2009

nonn

Previous name was: a(n) is the n-th A_0-prime (Archimedes_0 prime).

We have: (1) N is A_0-prime if and only if N is even, p = 2N + 1 is a prime number and both +2 and -2 generate Z_p^* (the multiplicative group of Z_p); (2) N is A_0-prime if and only if N = 2 (mod 4), p = 2N + 1 is a prime number and both +2 and -2 generate Z_p^*.

P. R. J. Asveld, Table of n, a(n) for n = 1..3378.
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011), 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. Another link.
Index entries for sequences related to the Josephus Problem

The A_0-primes are the even T- or Twist-primes, these T-primes are equal to the Queneau-numbers (A054639). For the related A_1-, A^+_1- and A^-_1-primes, see A163778, A163779 and A163780. Considered as sets A163777 is the intersection of the Josephus_2-primes (A163782) and the dual Josephus_2-primes (A163781), it also equals the difference of A054639 and the A_1-primes (A163779).

Cf. A137310.

okQ[n_] := EvenQ[n] && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* Jean-François Alcover, Sep 10 2019, from PARI *)

Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
ok(n)={n>1 && n==Follow(1, j->ceil((n+1)/2) - (-1)^j*ceil((j-1)/2))}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

ok(n)={n%2==0 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == 2*n}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(n) = 2*A137310(n). - Andrew Howroyd, Nov 11 2017

Definition simplified by Michel Marcus, May 27 2013
a(33)-a(55) from Andrew Howroyd, Nov 11 2017
New name from Joerg Arndt, Mar 23 2018, edited by M. F. Hasler, Mar 24 2018

A163781 a(n) is the n-th dJ_2 prime (dual Josephus_2 prime).

Original entry on oeis.org

2, 3, 6, 11, 14, 18, 23, 26, 30, 35, 39, 50, 51, 74, 83, 86, 90, 95, 98, 99, 119, 131, 134, 135, 146, 155, 158, 174, 179, 183, 186, 191, 194, 210, 230, 231, 239, 243, 251, 254, 270, 278, 299, 303, 306, 323, 326, 330, 338, 350, 354, 359, 371, 375, 378

Offset: 1

Peter R. J. Asveld, Aug 17 2009

nonn

The family of dual Josephus_2 (or dJ_2) permutations is defined by p(m,N)=(2N + 1 - F(m,2N + 1))/2 if 1<=m<=N, N>=2, where F(x,y) is the odd number such that 1<=F(x,y)=0. Note that F(2k + 1,y)=2k + 1 for 2k + 1

dJ_2 permutations can also be defined using a numbering/elimination procedure similar to the definition of the Josephus_2 permutations in [R.L. Graham et al.], or in A163782; see [P. R. J. Asveld].

No formula is known for a(n): the dJ_2 primes have been found by exhaustive search. But we have: (1) N is dJ_2 prime iff p=2N+1 is a prime number and -2 generates Z_p^* (the multiplicative group of Z_p); (2) N is dJ_2 prime iff p=2N+1 is a prime number and exactly one of the following holds:

(a) N=2 (mod 4) and both +2 and -2 generate Z_p^*,

(b) N=3 (mod 4) and -2 generates Z_p^* but +2 does not.

			For N=6 we have
  m       | 1   2   3   4   5   6
  --------+----------------------
  F(m,13) | 1   7   3  11   5   9
  t       | 0   2   0   1   0   3
  p(m,6)  | 6   3   5   1   4   2
So the permutation is (1 6 2 3 5 4) and 6 is dJ_2 prime.

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

P. R. J. Asveld, Table of n, a(n) for n=1..6756
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. University link.
Index entries for sequences related to the Josephus Problem

Considered as sets the union of A163781 and A163782 (J_2 primes) equals A054639 (T-primes or Queneau numbers), their intersection is equal to A163777 (Archimedes_0 primes). A163781 equals the union of A163777 and A163780 (Archimedes^-_1 primes).

okQ[n_] := Mod[n, 4] >= 2 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == If[OddQ[n], n, 2n];
Select[Range[1000], okQ] (* Jean-François Alcover, Sep 23 2019, from PARI *)

ok(n)={n%4>=2 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%2,n,2*n)};
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(37)-a(55) from Andrew Howroyd, Nov 11 2017

A163776 a(n) is the n-th dS-prime (dual Shuffle prime).

Original entry on oeis.org

4, 6, 12, 22, 28, 36, 46, 52, 60, 70, 78, 100, 102, 148, 166, 172, 180, 190, 196, 198, 238, 262, 268, 270, 292, 310, 316, 348, 358, 366, 372, 382, 388, 420, 460, 462, 478, 486, 502, 508, 540, 556, 598, 606, 612, 646, 652, 660, 676, 700, 708, 718, 742, 750, 756

Offset: 1

Peter R. J. Asveld, Aug 13 2009

nonn

For N>=2, the family of dual shuffle permutations is defined by p(m,N) = -2m (mod N+1) if N is even, p(m,N) = -2m (mod N) if N is odd and 1<=m

No formula is known for a(n): the dS-primes have been found by exhaustive search. But we have: N is dS-prime iff p=N+1 is an odd prime number and -2 generates Z_p^* (the multiplicative group of Z_p).

			For N=6 and N=10 we obtain the permutations (1 5 4 6 2 3) and (1 9 4 3 5)(2 7 8 6 10): 6 is dS-prime, but 10 is not.

P. R. J. Asveld, Table of n, a(n) for n=1..3612
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. University link.
Index entries for sequences related to the Josephus Problem

a(n)/2 results in the dual Josephus_2-primes (A163781). Considered as sets a(n)/2 is the union of A163777 and A163780. If b(n) denotes the shuffle primes (A071642), then the union of a(n)/2 and b(n)/2 is equal to the Twist-primes or Queneau numbers (A054639), their intersection is equal to the Archimedes_0-primes (A163777).

a(n) = 2*A163781(n).

a(33)-a(55) from Andrew Howroyd, Nov 11 2017

A163783 a(n) is the n-th J_3-prime (Josephus_3 prime).

Original entry on oeis.org

3, 5, 27, 89, 1139, 1219, 1921, 2155, 5775, 9047, 12437, 78785, 105909, 197559, 1062657

Offset: 1

Peter R. J. Asveld, Aug 05 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 3rd unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_3-prime if this permutation consists of a single cycle of length N.
There are 14 J_3-primes in the interval 2..1000000 only. No formula is known; the J_3-primes have been found by exhaustive search.
a(16) > 3*10^6. - Bert Dobbelaere, Apr 20 2019

			All J_3-primes are odd.

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology.
P. R. J. Asveld, Permuting Operations on Strings and Their Relation to Prime Numbers Discrete Applied Mathematics 159 (2011) 1915-1932.
Index entries for sequences related to the Josephus Problem

A163782 for J_2-primes. A163784 through A163800 for J_4- through J_20-primes.

a(15) from Bert Dobbelaere, Apr 20 2019

A163784 a(n) is the n-th J_4-prime (Josephus_4 prime).

Original entry on oeis.org

2, 5, 10, 369, 609, 1841, 2462, 3297, 3837, 14945, 94590, 98121, 965013, 1634157

Offset: 1

Peter R. J. Asveld, Aug 05 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 4th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_4-prime if this permutation consists of a single cycle of length N.
There are 13 J_4-primes in the interval 2..1000000 only. No formula is known; the J_4-primes have been found by exhaustive search.
a(15) > 3*10^6. - Bert Dobbelaere, Apr 20 2019

			2 is a J_4-prime (trivial).

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology.
P. R. J. Asveld, Permuting Operations on Strings and Their Relation to Prime Numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
Index entries for sequences related to the Josephus Problem

A163782 through A163783 for J_2- through J_3-primes. A163785 through A163800 for J_5- through J_20-primes.

a(14) from Bert Dobbelaere, Apr 20 2019

A163785 a(n) is the n-th J_5-prime (Josephus_5 prime).

Original entry on oeis.org

3, 15, 17, 45, 73, 83, 165, 177, 181, 229, 377, 383, 787, 2585, 3127, 3635, 4777, 36417, 63337, 166705, 418411

Offset: 1

Peter R. J. Asveld, Aug 05 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 5th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_5-prime if this permutation consists of a single cycle of length N.

There are 21 J_5-primes in the interval 2..1000000 only. No formula is known; the J_5-primes have been found by exhaustive search.

			All J_5-primes are odd.

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

P. R. J. Asveld, Permuting Operations on Strings and Their Relation to Prime Numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. University link.
Index entries for sequences related to the Josephus Problem

Cf. A163782 through A163784 for J_2- through J_4-primes.

Cf. A163786 through A163800 for J_6- through J_20-primes.

A163786 a(n) is the n-th J_6-prime (Josephus_6 prime).

Original entry on oeis.org

2, 13, 17, 18, 34, 49, 93, 97, 106, 225, 401, 745, 2506, 3037, 3370, 4713, 5206, 8585, 13418, 32237, 46321, 75525, 97889, 106193, 238513, 250657, 401902, 490118

Offset: 1

Peter R. J. Asveld, Aug 05 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 6th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_6-prime if this permutation consists of a single cycle of length N.

There are 28 J_6-primes in the interval 2..1000000 only. No formula is known; the J_6-primes were found by exhaustive search.

			2 is a J_6-prime (trivial).

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

P. R. J. Asveld, Permuting Operations on Strings and Their Relation to Prime Numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. University link.
Index entries for sequences related to the Josephus Problem

Cf. A163782 through A163785 for J_2- through J_5-primes.

Cf. A163787 through A163800 for J_7- through J_20-primes.

cp's OEIS Frontend

User: Peter R. J. Asveld

A163780 Terms in A054639 equal to 3 mod 4.

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A163778 Odd terms in A054639.

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A163779 Numbers k of the form 4*j + 1 such that 2*k + 1 is a prime with primitive root 2.

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A163777 Even terms in the sequence of Queneau numbers A054639.

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A163781 a(n) is the n-th dJ_2 prime (dual Josephus_2 prime).

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A163776 a(n) is the n-th dS-prime (dual Shuffle prime).

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A163783 a(n) is the n-th J_3-prime (Josephus_3 prime).

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A163779 Numbers k of the form 4j + 1 such that 2k + 1 is a prime with primitive root 2.