Asher Gray - cp's OEIS frontend

A380827 Least integer k such that the multiplicative group modulo n is a subgroup of the symmetric group S_k.

1, 1, 2, 2, 4, 2, 5, 4, 5, 4, 7, 4, 7, 5, 6, 6, 16, 5, 11, 6, 7, 7, 13, 6, 9, 7, 11, 7, 11, 6, 10, 10, 9, 16, 9, 7, 13, 11, 9, 8, 13, 7, 12, 9, 9, 13, 25, 8, 12, 9, 18, 9, 17, 11, 11, 9, 13, 11, 31, 8, 12, 10, 10, 18, 11, 9, 16, 18, 15, 9, 14, 9, 17, 13, 11, 13, 12, 9, 18, 10, 29

Offset: 1

Asher Gray, Feb 04 2025

nonn

Asher Gray, Table of n, a(n) for n = 1..10000
Wikipedia, Multiplicative group of integers modulo n

Cf. A008475, A380222, A282625.

a(j*k) = a(j) + a(k) where j and k are coprime and both greater than 2.
a(2j) = j where j is odd.
a(2) = 1, a(4) = 2, a(2^k) = 2 + 2^(k-2) for k >= 3.
a(p^k) = A008475(p-1) * p^(k-1) for odd prime p.

A380222 Highest integer k such that the multiplicative group modulo k is a subgroup of the symmetric group S_n.

Original entry on oeis.org

2, 6, 6, 12, 18, 30, 42, 60, 90, 126, 210, 252, 420, 630, 840, 1260, 1680, 2730, 3276, 5460, 8190, 10920, 16380, 21840, 32760, 40950, 65520, 90090, 120120, 180180, 253890, 360360, 507780, 720720, 1015560, 1332240, 2031120, 2792790, 3996720, 5585580

Offset: 1

Asher Gray, Jan 17 2025

nonn

a(n) is the highest k for which A380827(k) <= n.

			a(2) = 6 because (Z/6Z)* is a subgroup of S_2 (isomorphic to it in fact) and there is no modulus k with k > 6 and (Z/kZ)* a subgroup of S_2.

Asher Gray, Table of n, a(n) for n = 1..5000
Asher Gray, Highest modulus within S_n, Github repository.

Cf. A380827, A034891.

A379424 Least modulus k such that the multiplicative group modulo k has a difference of n nontrivial cycles between its minimal and maximal representation.

Original entry on oeis.org

1, 7, 31, 211, 1333, 6541, 45787, 281263, 1968841, 13781887, 93098053, 649998793, 4549991551, 31849940857, 215149600483, 1506047203381, 10542330423667, 86982188480467, 587573558919073, 4113014912433511, 28791104387034577, 247368468304929733

Offset: 0

Asher Gray, Dec 22 2024

nonn

This is equal to the least modulus k such that (Z/kZ)* has a representation as a direct product of cyclic groups, of which n are odd cycles. The number of even cycles in the maximal representation is equal to the total cycles in the minimal representation.

			a(4) = 1333 because (Z/1333Z) ≅ C210 x C6 ≅ C2 x C3 x C5 x C2 x C3 x C7. The first representation has 2 cycles and the second has 6, a difference of 4.

Asher Gray, Table of n, a(n) for n = 0..500
Asher Gray, Least modulus with n cycles, Github repository.
Asher Gray, Sequences from Group Theory, YouTube Video.

Cf. A102476, A379423.

A379423 Least modulus k such that the multiplicative group modulo k is the direct product of n nontrivial cyclic groups.

Original entry on oeis.org

1, 3, 7, 21, 56, 168, 504, 1736, 5208, 15624, 57288, 171864, 671832, 2234232, 7390152, 32023992, 96071976, 450799272, 1559322072, 5860390536, 20271186936, 95118646392, 385152551784, 1236542403096, 6182712015480, 23494305658824, 82848341007432, 409295535424776

Offset: 0

Asher Gray, Dec 22 2024

nonn

Compare with A102476. That sequence also measures the least modulus k with n nontrivial cyclic groups, but only using the rank, the minimal representation for each such k. For example, A102476(3) = 24 as (Z/24Z)* ≅ C2 x C2 x C2. However with this sequence a(3) = 21 as (Z/21Z)* ≅ C2 x C2 x C3.

			a(2) = 7 because (Z/7Z)* ≅ C2 x C3.

Asher Gray, Table of n, a(n) for n = 0..1000
Asher Gray, Least modulus with n cycles, Github repository.
Asher Gray, Sequences from Group Theory, YouTube Video.

Cf. A102476.

cp's OEIS Frontend

User: Asher Gray

A380827 Least integer k such that the multiplicative group modulo n is a subgroup of the symmetric group S_k.

Views

Author

Keywords

Links

Crossrefs

Formula

A380222 Highest integer k such that the multiplicative group modulo k is a subgroup of the symmetric group S_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A379424 Least modulus k such that the multiplicative group modulo k has a difference of n nontrivial cycles between its minimal and maximal representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A379423 Least modulus k such that the multiplicative group modulo k is the direct product of n nontrivial cyclic groups.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs