A364186 Primes p such that p divides 2^((p-1)/x) - 1, where x is the smallest odd prime factor of p - 1.

31, 43, 109, 127, 157, 223, 229, 251, 277, 283, 307, 397, 431, 433, 439, 457, 499, 601, 641, 643, 691, 727, 733, 739, 811, 911, 919, 953, 971, 997, 1013, 1021, 1051, 1069, 1093, 1103, 1163, 1181, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1709

Offset: 1

Author

Arkadiusz Wesolowski, Jul 15 2023

Keywords

• nonn

References

• Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Crossrefs

Supersequence of A014752.

Cf. A059858, A270802, A360652.

Programs

Magma

[p: p in PrimesUpTo(1709) | #Factorization(p-1) ge 2 and Modexp(2, Truncate((p-1)/Factorization(p-1)[2][1]), p) eq 1];

PARI

forprime(p=2, 1709, v=Vec(factor(p-1))[1]; if(#v>1, t=0; e=v[2]; x=floor(p^(1/e))+1; until(x==p||t==2, if(Mod(x, p)^e==2, t++); x++); if(t==2, print1(p, ", "))));

PARI

isok(p) = my(v=Vec(factor(p-1))[1]); isprime(p) && #v>1 && Mod(2, p)^((p-1)/v[2])==1;