A347332 - OEIS

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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..5.

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.

EXAMPLE

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.

PROG

(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

CROSSREFS

Cf. A005385, A059456.

Sequence in context: A128348 A029973 A344736 * A054551 A049065 A274609

Adjacent sequences: A347329 A347330 A347331 * A347333 A347334 A347335

KEYWORD

nonn,hard,more

AUTHOR

Tomohiro Yamada, Aug 27 2021

STATUS

approved