OFFSET
1,1
COMMENTS
In other words, odd k>1 such that there are no bases 2 <= b <= k-2 in which k is an Euler-Jacobi pseudoprime.
Odd k = 2*m+1 such that A329726(m) = 2.
According to the formula of A329726, k is a term if and only if k falls into one of these categories:
(a) k == 3 (mod 4), and gcd(p-1,k-1) = 2 for every prime factor p of k;
(b) k is a power of 3;
(c) k = p^a*q^b, p == q == 3 (mod 4), a and b odd, and gcd(p-1,q^b-1) = gcd(p^a-1,q-1) = 2.
See A329726 for a more general statement.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
399 = 3*7*19 is a term since 399 == 3 (mod 4), and we have gcd(3-1,399-1) = gcd(7-1,399-1) = gcd(19-1,399-1) = 2.
231 = 3*7*11 is not a term since gcd(11-1,231-1) != 2. Indeed, A329726((231-1)/2) = 10.
33 = 3*11 is a term since 3 == 11 == 3 (mod 4), and gcd(3-1,11-1) = 2.
35937 = 3^3*11^3 is a term since gcd(3^3-1,11-1) = gcd(3-1,11^3-1) = 2.
5589 = 23*3^5 is not a term since gcd(23-1,3^5-1) != 2. Indeed, A329726((5589-1)/2) = 22.
16281 = 67*3^5 is not a term since gcd(67-1,3^5-1) != 2. Indeed, A329726((16281-1)/2) = 22.
2133 = 79*3^3 is not a term since gcd(79-1,3^3-1) != 2. Indeed, A329726((2133-1)/2) = 26.
3537 = 131*3^3 is not a term since gcd(131-1,3^3-1) != 2. Indeed, A329726((3537-1)/2) = 26.
PROG
(PARI) isA394182(k) = if(k%2==0, return(0)); if(k>1 && isprimepower(3*k), return(1)); my(f=factor(k), r=#f~); if(k%4==3, for(i=1, r, if(gcd(f[i, 1]-1, k-1)>2, return(0))); return(1)); return(r==2 && f[1, 1]%4==3 && f[1, 2]%2==1 && f[2, 1]%4==3 && f[2, 2]%2==1 && gcd(f[1, 1]-1, k-1) == 2 && gcd(f[2, 1]-1, k-1) == 2)
CROSSREFS
A394177, which is the sequence where "Euler-Jacobi pseudoprime" is replaced by "strong pseudoprime", is a subsequence.
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Mar 11 2026
STATUS
approved