A305333 -id:A305333 - cp's OEIS frontend

A305331 Multiplicative order of 5 (mod p^2), where p = prime(n), or 0 if 5 and p are not coprime.

1, 6, 0, 42, 55, 52, 272, 171, 506, 406, 93, 1332, 820, 1806, 2162, 2756, 1711, 1830, 1474, 355, 5256, 3081, 6806, 3916, 9312, 2525, 10506, 11342, 2943, 12656, 5334, 8515, 18632, 9591, 5513, 11325, 24492, 8802, 27722, 29756, 15931, 2715, 3629, 37056, 38612

Offset: 1

Felix Fröhlich, May 30 2018

nonn

Cf. A211241, A305332, A305333.

Table[If[p==5, 0, MultiplicativeOrder[5, p^2]], {p, Prime@ Range@ 45}] (* Giovanni Resta, May 31 2018 *)

a(n) = my(p=prime(n)); if(p==5, return(0), return(znorder(Mod(5, p^2))))

A305332 Multiplicative order of 5 (mod A123692(n)^2).

Original entry on oeis.org

1, 10385, 40486, 13367790, 1645333506, 6692367336, 11796759175

Offset: 1

Felix Fröhlich, May 30 2018

From Eric Chen, Jun 07 2018: (Start)
    b    known Wieferich primes in base b (multiplicative order of b mod these primes (also these primes^2)) (if the order is p-1, then b is a primitive root to mod this prime (but not mod this prime^2), see A055578)
  1093 (364), 3511 (1755)
  11 (5), 1006003 (1006002)
  1093 (182), 3511 (1755)
  2 (1), 20771 (10385), 40487 (40486), 53471161 (13367790), 1645333507 (1645333506), 6692367337 (6692367336), 188748146801 (11796759175)
  66161 (66160), 534851 (106970), 3152573 (788143)
  5 (4), 491531 (245765)
  3 (2), 1093 (364), 3511 (585)
  2 (1), 11 (5), 1006003 (503001)
  3 (1), 487 (486), 56598313 (56598312)
  71 (70)
  2693 (2692), 123653 (123652)
  2 (1), 863 (862), 1747591 (873795)
  29 (28), 353 (352), 7596952219 (7596952218)
  29131 (29130), 119327070011 (59663535005)
  1093 (91), 3511 (1755)
  2 (1), 3 (2), 46021 (7670), 48947 (24473), 478225523351 (478225523350)
  5 (4), 7 (3), 37 (36), 331 (110), 33923 (33922), 1284043 (428014)
  3 (1), 7 (6), 13 (12), 43 (42), 137 (68), 63061489 (63061488)
  281 (140), 46457 (46456), 9377747 (9377746), 122959073 (122959072)
  2 (1)
  13 (3), 673 (224), 1595813 (797906), 492366587 (246183293), 9809862296159 (44999368331)
  13 (6), 2481757 (827252), 13703077 (13703076), 15546404183 (7773202091), 2549536629329 (2549536629328)
  5 (2), 25633 (6408)
These orders n will satisfy that Phi_n(b) is divisible by p^2, where Phi is the cyclotomic polynomial. (Usually, Phi_n(b) is squarefree, but these are all exceptions; i.e., if p^2 divides Phi_n(b) (except the case p = 2, n = 2 and b == 3 (mod 4)), then p is a Wieferich prime in base b.)
(End)

Cf. A123692, A211241, A305331, A305333.

v=[2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801]; for(k=1, #v, print1(znorder(Mod(5, v[k]^2)), ", "))

a(n) = A305331(A123692(n)).

cp's OEIS Frontend

A305331 Multiplicative order of 5 (mod p^2), where p = prime(n), or 0 if 5 and p are not coprime.

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