A328414 Numbers k such that (Z/mZ)* = C_2 X C_(2k) has no solutions m, where (Z/mZ)* is the multiplicative group of integers modulo m.

7, 12, 13, 17, 19, 25, 28, 31, 34, 37, 38, 43, 47, 49, 52, 57, 59, 61, 62, 67, 71, 73, 76, 77, 79, 80, 84, 85, 91, 92, 93, 94, 97, 100, 101, 103, 104, 107, 108, 109, 112, 117, 118, 121, 122, 124, 127, 129, 133, 137, 139, 142, 143, 144, 148, 149, 151, 152, 157, 160, 161, 163, 164

Offset

1,1

Comments

Indices of 0 in A328410, A328411 and A328412.

By definition, if there is no such m that psi(m) = 2k, psi = A002322, then m is a term of this sequence.

Links

Jianing Song, Table of n, a(n) for n = 1..10001

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Wikipedia, Multiplicative group of integers modulo n.

Example

12 is a term: if there exists m such that (Z/mZ)* = C_2 X C_24 = C_2 X C_8 X C_3, then m must have a factor q such that q is an odd prime power and phi(q) = 8 or phi(q) = 24, phi = A000010, which is impossible.
80 is a term: if there exists m such that (Z/mZ)* = C_2 X C_80 = C_2 X C_16 X C_5, then m must have a factor q such that q is an odd prime power and phi(q) = 80 or phi(q) = 16, which is impossible.

Prog

(PARI) isA328414(n) = my(r=4*n, N=floor(exp(Euler)*r*log(log(r^2))+2.5*r/log(log(r^2)))); for(k=r+1, N+1, if(eulerphi(k)==r && lcm(znstar(k)[2])==r/2, return(0)); if(k==N+1, return(1)))
for(n=1, 200, if(isA328414(n), print1(n, ", ")))
(PARI) isA328414(n) = (#select(x->znstar(x)[2]==[2*n, 2], invphi(4*n)) == 0) \\ Jianing Song, Apr 08 2026 using Max Alekseyev's invphi.gp

Crossrefs

Cf. A328410, A328411, A328412. Complement of A328413.

Cf. also A000010, A002322, A005277, A079695.

Sequence in context: A070420 A223423 A274334 * A083681 A178660 A048653

Adjacent sequences: A328411 A328412 A328413 * A328415 A328416 A328417

Keyword

nonn

Author

Jianing Song, Oct 14 2019

Status

approved