A394179 Least Carmichael number (A002997) that is the product of exactly n primes congruent to 3 modulo 4.

8911, 1773289, 1419339691, 4077957961, 3475350807391, 440515336876021, 574539328092938671, 2426698123549677901, 4971170854788923506051, 447301514144395660522501, 297982503152489384707421251

Offset

3,1

Comments

If k is an odd number with at least 3 distinct prime factors, then the number of bases in (Z/kZ)^* in which k is a strong pseudoprime at most phi(k)/2^(omega(k)-1), where equality holds if and only if k is a Carmichael number with only prime factors congruent to 3 modulo 4 (A329468). See A329759 for a proof.

Data copied from A329468.

Links

Table of n, a(n) for n=3..13.

Wikipedia, Strong pseudoprime

Index entries for sequences related to pseudoprimes

Prog

(PARI)
carmichael_3mod4(A, B, k) = A=max(A, vecprod(primes(k+1))\2); local f; (f = (m, l, lo, k) -> my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(1/Mod(m, l))); while(t < lo, t += l); forstep(p=t, hi, l, if(p%4 == 3 && (m*p-1) % (p-1) == 0 && isprime(p), listput(list, m*p))), forprime(p=lo, hi, if(p%4 == 3 && gcd(m, p-1) == 1, list=concat(list, f(m*p, lcm(l, p-1), p+1, k-1))))); list); vecsort(Vec(f(1, 1, 3, k)));
a(n) = if(n < 3, return()); my(x=1, y=2*x); while(1, my(v=carmichael_3mod4(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Mar 17 2026

Crossrefs

Cf. A329468, A071294, A329759.

Sequence in context: A329468 A185321 A267462 * A256237 A065235 A087167

Adjacent sequences: A394176 A394177 A394178 * A394180 A394181 A394182

Keyword

nonn,hard,more

Author

Jianing Song, Mar 11 2026

Extensions

a(11)-a(12) from Amiram Eldar, Mar 13 2026

a(13) from Daniel Suteu, Mar 17 2026

Status

approved