A387182 Positive integers k where, for zeta_k a primitive k-th root of unity, the set S_k = {(1-zeta_k^i)/(1-zeta_k) : 1<i<k/2, GCD(i,k)=1} has a nontrivial (multiplicative) dependence relation.

39, 55, 56, 65, 68, 78, 91, 95, 105, 110, 111, 112, 117, 120, 123, 124, 130, 132, 133, 136, 140, 145, 153, 155, 156, 164, 165, 168, 171, 172, 175, 182, 183, 184, 190, 195, 201, 203, 205, 209, 210, 219, 220, 221, 222, 224, 228, 234, 240, 246, 248, 253, 255, 259

Offset

1,1

Comments

By Theorem 3(2) in Feng's paper, the existence of a dependence relation in S can be shown for k!=2 mod 4 as follows. For k with prime factorization k=p_1^e_1*...*p_l^e_l, let k_i=k/(p_i^e_i) for i=1,...,l. If there is some i for which, in the group of units mod k_i, the subgroup generated by -1 and p_i does not generate the entire unit group mod k_i, then k is in this sequence.

It follows that no primes or prime powers are in this sequence.

If k=2 mod 4, let k'=k/2. The k'-th and k-th cyclotomic fields are equal. There is a dependence relation in S_k if and only if there is a dependence relation in S_k'. As a result, if k=2 mod 4, then k is in this sequence if and only if k/2 is in this sequence.

It is known that if k is in this sequence, then all multiples of k are in this sequence as well.

It is known that if k!=2 mod 4 and k has at least 4 prime divisors, then k is in this sequence.

Links

Table of n, a(n) for n=1..54.

Keqin Feng, The Rank of Group of Cyclotomic Units in Abelian Fields, Journal of Number Theory, 14 (1982), 315-326.

Caleb M. Shor and Jae Hyung Sim, Equidistribution Conditions for Gaps of Geometric Numerical Semigroups, arXiv:2503.10826 [math.NT], 2025.

Example

Let k=39. Then k=3^1*13^1. Then k_1=39/3=13 and k_2=39/13=3. For U_13 the group of units mod 13, the subgroup of U_13 generated by -1 and 3 is <-1,3>={1,3,4,9,10,12}. Since <-1,3>!=U_13, k=39 is in this sequence.
For k=39, a dependence relation among elements of S_k is Product_{g in {4,10,14,16,17}} c_g^78 * Product_{h in {2,5,7,8,11,19}} c_h^(-78) = 1, where c_j=(1-zeta_39^j)/(1-zeta_39). (See Section 3.3 of Shor and Sim's paper for details.)
For a non-example, let k=21. Working in U_7, <-1,3>=U_7. And working in U_3, <-1,7>=U_3. Hence k=21 is not in this sequence.

Crossrefs

Sequence in context: A133676 A317987 A330219 * A013658 A227735 A319983

Adjacent sequences: A387179 A387180 A387181 * A387183 A387184 A387185

Keyword

nonn

Author

Caleb M. Shor, Sep 28 2025

Status

approved