A163800 -id:A163800 - cp's OEIS frontend

A163792 a(n) is the n-th J_12-prime (Josephus_12 prime).

2, 38, 57, 145, 189, 2293, 2898, 6222, 7486, 26793, 45350, 90822, 177773

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 12th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_12-prime if this permutation consists of a single cycle of length N.

There are 13 J_12-primes in the interval 2..1000000 only. No formula is known; the J_12-primes were found by exhaustive search.

			2 is a J_12-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 A163791 for J_2- through J_11-primes.

Cf. A163793 through A163800 for J_13- through J_20-primes.

A163793 a(n) is the n-th J_13-prime (Josephus_13 prime).

Original entry on oeis.org

5, 57, 117, 187, 251, 273, 275, 665, 2511, 40393, 48615, 755921, 970037

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 13th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_13-prime if this permutation consists of a single cycle of length N.

There are 13 J_13-primes in the interval 2..1000000 only. No formula is known; the J_13-primes have been found by exhaustive search.

			All J_13-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 A163792 for J_2- through J_12-primes.

Cf. A163794 through A163800 for J_14- through J_20-primes.

A163794 a(n) is the n-th J_14-prime (Josephus_14 prime).

Original entry on oeis.org

2, 185, 205, 877, 2045, 3454, 6061, 29177, 928954

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 14th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_14-prime if this permutation consists of a single cycle of length N.

There are 9 J_14-primes in the interval 2..1000000 only. No formula is known; the J_14-primes were found by exhaustive search.

			2 is a J_14-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 A163793 for J_2- through J_13-primes.

Cf. A163795 through A163800 for J_15- through J_20-primes

A163795 a(n) is the n-th J_15-prime (Josephus_15 prime).

Original entry on oeis.org

3, 9, 13, 25, 49, 361, 961, 1007, 2029, 8593, 24361, 44795, 88713

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 15th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_15-prime if this permutation consists of a single cycle of length N.

There are 13 J_15-primes in the interval 2..1000000 only. No formula is known; the J_15-primes have been found by exhaustive search.

			All J_15-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 A163794 for J_2- through J_14-primes.

Cf. A163796 through A163800 for J_16- through J_20-primes.

A163796 a(n) is the n-th J_16-prime (Josephus_16 prime).

Original entry on oeis.org

2, 14, 49, 333, 534, 550, 2390, 3682, 146794, 275530, 687245, 855382, 2827062, 3062118, 3805189

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 16th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_16-prime if this permutation consists of a single cycle of length N.

There are 12 J_16-primes in the interval 2..1000000 only. No formula is known; the J_16-primes were found by exhaustive search.

			2 is a J_16-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 A163795 for J_2- through J_15-primes.

Cf. A163797 through A163800 for J_17- through J_20-primes.

a(13)-a(15) from Jinyuan Wang, Jul 03 2025

A163797 a(n) is the n-th J_17-prime (Josephus_17 prime).

Original entry on oeis.org

3, 5, 7, 39, 93, 267, 557, 2389, 2467, 4059, 4681, 6213, 70507, 151013, 282477, 421135, 1272901

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 17th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_17-prime if this permutation consists of a single cycle of length N.

There are 16 J_17-primes in the interval 2..1000000 only. No formula is known; the J_17-primes have been found by exhaustive search.

			All J_17-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 A163796 for J_2- through J_16-primes.

Cf. A163798 through A163800 for J_18- through J_20-primes.

a(17) from Jinyuan Wang, Jul 05 2025

A163798 a(n) is the n-th J_18-prime (Josephus_18 prime).

Original entry on oeis.org

2, 5, 462, 530, 6021, 14686, 19537, 67161, 1766014, 1921621, 2779501

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 18th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_18-prime if this permutation consists of a single cycle of length N.

There are 8 J_18-primes in the interval 2..1000000 only. No formula is known; the J_18-primes were found by exhaustive search.

			2 is a J_18-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 A163797 for J_2- through J_17-primes.

Cf. A163799 through A163800 for J_19- through J_20-primes.

a(9)-a(11) from Jinyuan Wang, Jul 05 2025

A163799 a(n) is the n-th J_19-prime (Josephus_19 prime).

Original entry on oeis.org

15, 145, 149, 243, 259, 449, 1921, 2787, 15871, 18563, 26459, 191515, 283269, 741343, 844805

Offset: 1

Peter R. J. Asveld, Aug 04 2009

Place the numbers 1..N (N>=2) on a circle and cyclicly mark the 19th unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_19-prime if this permutation consists of a single cycle of length N.

There are 15 J_19-primes in the interval 2..1000000 only. No formula is known; the J_19-primes have been found by exhaustive search.

			All J_19-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 A163798 and A163800 for J_2- through J_18- and J_20-primes.

A357217 Array read by descending antidiagonals: T(n,k) is the number of cycles of the permutation given by the order of elimination in the Josephus problem for n numbers and a count of k; n, k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 5, 1, 2, 2, 2, 1, 6, 1, 1, 1, 2, 1, 1, 7, 1, 2, 2, 2, 1, 2, 4, 8, 1, 1, 3, 2, 3, 3, 3, 2, 9, 1, 2, 2, 2, 3, 2, 2, 2, 1, 10, 1, 1, 1, 2, 1, 3, 3, 2, 3, 5, 11, 1, 2, 2, 2, 3, 2, 2, 4, 5, 2, 2, 12, 1, 1, 1, 2, 3, 1, 3, 2, 3, 1, 3, 2, 13

Offset: 1

Pontus von Brömssen, Sep 18 2022

n >= 2 is a Josephus_k prime if and only if T(n,k) = 1; see A163782-A163800.

			Array begins:
  n\k|  1  2  3  4  5  6  7  8  9 10
  ---+------------------------------
   1 |  1  1  1  1  1  1  1  1  1  1
   2 |  2  1  2  1  2  1  2  1  2  1
   3 |  3  2  1  2  1  2  3  2  1  2
   4 |  4  2  2  2  2  2  2  2  2  2
   5 |  5  1  1  1  3  3  1  3  3  3
   6 |  6  1  2  3  2  3  2  1  2  3
   7 |  7  4  3  2  3  2  3  2  5  2
   8 |  8  2  2  2  4  2  2  4  6  2
   9 |  9  1  3  5  3  3  3  3  3  3
  10 | 10  5  2  1  2  3  2  1  2  3
For n = 4, k = 2, the order of elimination is (2,4,3,1) (row 4 of A321298). This permutation has two cycles, (1 2 4) and (3), so T(4,2) = 2.

Pontus von Brömssen, Antidiagonals n = 1..100, flattened
James Dowdy and Michael E. Mays, Josephus permutations, Journal of Combinatorial Mathematics and Combinatorial Computing 6 (1989), 125-130.
Wikipedia, Josephus problem
Index entries for sequences related to the Josephus Problem

Cf. A003418, A006694 (column k=2), A163782-A163800 (Josephus primes), A198789, A321298 (the Josephus permutations for k=2).

from sympy.combinatorics import Permutation
def A357217(n,k):
    return Permutation.josephus(k,n).cycles

T(n,k+A003418(n)) = T(n,k), i.e., the n-th row is periodic with period A003418(n).

cp's OEIS Frontend

A163792 a(n) is the n-th J_12-prime (Josephus_12 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A163793 a(n) is the n-th J_13-prime (Josephus_13 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A163794 a(n) is the n-th J_14-prime (Josephus_14 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A163795 a(n) is the n-th J_15-prime (Josephus_15 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A163796 a(n) is the n-th J_16-prime (Josephus_16 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A163797 a(n) is the n-th J_17-prime (Josephus_17 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A163798 a(n) is the n-th J_18-prime (Josephus_18 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A163799 a(n) is the n-th J_19-prime (Josephus_19 prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A357217 Array read by descending antidiagonals: T(n,k) is the number of cycles of the permutation given by the order of elimination in the Josephus problem for n numbers and a count of k; n, k >= 1.

Views

Author

Keywords