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A305332 Multiplicative order of 5 (mod A123692(n)^2). 2
1, 10385, 40486, 13367790, 1645333506, 6692367336, 11796759175 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)

    2    1093 (364), 3511 (1755)

    3    11 (5), 1006003 (1006002)

    4    1093 (182), 3511 (1755)

    5    2 (1), 20771 (10385), 40487 (40486), 53471161 (13367790), 1645333507 (1645333506), 6692367337 (6692367336), 188748146801 (11796759175)

    6    66161 (66160), 534851 (106970), 3152573 (788143)

    7    5 (4), 491531 (245765)

    8    3 (2), 1093 (364), 3511 (585)

    9    2 (1), 11 (5), 1006003 (503001)

   10    3 (1), 487 (486), 56598313 (56598312)

   11    71 (70)

   12    2693 (2692), 123653 (123652)

   13    2 (1), 863 (862), 1747591 (873795)

   14    29 (28), 353 (352), 7596952219 (7596952218)

   15    29131 (29130), 119327070011 (59663535005)

   16    1093 (91), 3511 (1755)

   17    2 (1), 3 (2), 46021 (7670), 48947 (24473), 478225523351 (478225523350)

   18    5 (4), 7 (3), 37 (36), 331 (110), 33923 (33922), 1284043 (428014)

   19    3 (1), 7 (6), 13 (12), 43 (42), 137 (68), 63061489 (63061488)

   20    281 (140), 46457 (46456), 9377747 (9377746), 122959073 (122959072)

   21    2 (1)

   22    13 (3), 673 (224), 1595813 (797906), 492366587 (246183293), 9809862296159 (44999368331)

   23    13 (6), 2481757 (827252), 13703077 (13703076), 15546404183 (7773202091), 2549536629329 (2549536629328)

   24    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)

LINKS

Table of n, a(n) for n=1..7.

FORMULA

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

PROG

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

CROSSREFS

Cf. A123692, A211241, A305331, A305333.

Sequence in context: A180460 A035912 A226599 * A271766 A104439 A290035

Adjacent sequences:  A305329 A305330 A305331 * A305333 A305334 A305335

KEYWORD

nonn,hard,more

AUTHOR

Felix Fröhlich, May 30 2018

STATUS

approved

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Last modified September 18 11:01 EDT 2021. Contains 347518 sequences. (Running on oeis4.)