A347332 Unsafe primes (primes in A059456) for which there is exactly one divisor d of p - 1 such that o(d) = L(p), where o(k) is the multiplicative order of k modulo p and L(p) is the least common multiple of o(k)'s among all divisors k of p - 1.

2, 3, 31, 43, 112643

Offset: 1

Author

Tomohiro Yamada, Aug 27 2021

Keywords

• nonn
• hard
• more

Comments

a(6) > 10^12, if it exists.

If a prime p = 2ab + 1 with a, b prime belongs to this sequence, then o(k) is a permutation over divisors of p - 1.

If a prime p = 2q + 1 is a safe prime (in A005385), then we have either o(2) = 2q, o(q) = q or o(2) = q, o(q) = 2q. Thus there is exactly one divisor d of p - 1 such that o(d) = L(p) = 2q.

Examples

			For p  = 31, then o(2) = 5, o(3) = 30, o(5) = 3, o(6) = 6, o(10) = 15, o(15) = 10, and o(30) = 2; so 31 is a term.
		

Links

• Peter Fletcher and Camron Withrow, Primes p Having at Most One Divisor of p-1 of a Specified Multiplicative Order, Integers 19 (2019), Article #A61. Only considers terms > 3.

Crossrefs

Cf. A005385, A059456.

Programs

PARI

isunsafe(p) = isprime(p) && !isprime(p\2); \\ A059456
isok(p) = {if (isunsafe(p), my(vd=divisors(p-1), L = lcm(vd)); sum(k=1, #vd, znorder(Mod(vd[k], p)) == L) == 1;);} \\ Michel Marcus, Aug 27 2021