primes p s.t. there is an A s.t. A^(p-1)==1 mod p^2

[Joerg Arndt]

seq. not in OEIS:

11, 29, 37, 43, 59, 71, 79, 97, 103, 109, 113, 127, 131, 137, 151,
163, 181, 191, 197, 199, 211, 223, 229, 233, 241, 257, 263, 269, 281,
283, 293, 307, 313, 331, 347, 349, 353, 359, 367, 373, 379, 397, 401,
419, 421, 433, 439, 449, 461, 463, 487, 499, 509, 521, 523, 547, 557,
563, 571, 577, 599, 601, 607, 617, 619, 631, 641, 647, 653, 659, 661,
673, 701, 727, 739, 743, 761, 769, 773, 797, 809, 823, 827, 829, 839,
857, 859, 863, 877, 883, 887, 907, 911, 919, 941, 947, 953, 967, 971,
977, 983, 991, 997,

Examples (pairs [p, a]):
[11, 3]
[11, 9]
[29, 14]
[37, 18]
[43, 19]
[59, 53]
[71, 11]
[71, 26]
[79, 31]
[97, 53]



{ forprime (p=2,1000,
    for (a=2,p-1,
        p2 = p^2;
        q = ( Mod(a,p2)^(p-1) == Mod(1,p2) );
        if ( q , print1(p,",");break() );
    );
 ); }

submit?
also submit complement?

btw. 113 is the only prime with A^(p-1)==1 mod p^3 I could
find;  none for modulus p^k where k>3 so far.