A125636 -id:A125636 - cp's OEIS frontend

A133862 Smallest odd prime base q such that p^12 divides q^(p-1) - 1, where p = prime(n).

12289, 1062881, 853235443, 92233439147, 3143820659087, 58713568184837, 2359162908109223, 2649283656602003, 53928980532177631, 557792163777408809, 2084452633098194627, 8958368398788306367, 15810453676175767201

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(12) = 12289.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133859, A133860, A133861, A133863, A133864, A133865.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^12 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

A133863 Smallest odd prime base q such that p^13 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

40961, 19131877, 2441406251, 115385868869, 138090848575723, 358661570404751, 44510586506850631, 252317900773542353, 4465433274456775633, 39171440762351329829, 11887418854442931407, 14582408526413537791

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(13) = 40961.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133859, A133860, A133861, A133862, A133864, A133865.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^13 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

A133864 Smallest odd prime base q such that p^14 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

65537, 19131877, 53834264557, 1356446145697, 488581592070877, 22771419458231473, 346100334752156863, 2467410166021233673, 19165875476832528551, 61879867860030528131, 1106827928513014993387

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(14) = 65537.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133859, A133860, A133861, A133862, A133863, A133865.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^14 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

A133865 Smallest odd prime base q such that p^15 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

65537, 57395627, 122070312499, 56020344873707, 6266190914259137, 65106791321062951, 12132548193910221893, 50407811312994280933, 172048888780798211059, 16668261908754510204233, 35965174106571679882189

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(15) = 65537.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133859, A133860, A133861, A133862, A133863, A133864.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^15 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

A249162 Square array A(p, b) read by antidiagonals in which rows are indexed by successive prime numbers p_i and row b(p_i) gives the smallest prime base b_n to which q = (p_i, b_(n-1)) is a Wieferich prime.

Original entry on oeis.org

2, 5, 3, 7, 17, 5, 19, 131, 7, 7, 127, 659, 19, 19, 11, 911, 503, 127, 127, 3, 13, 7331, 9833, 911, 911, 17, 19, 17, 167149, 49603, 7331, 7331, 131, 127, 131, 19, 387749, 327317, 167149, 167149, 659, 911, 659, 127, 23, 17153317, 13900147, 387749, 387749, 503

Offset: 1

Felix Fröhlich, Oct 22 2014

			A(6,4) = 911, since the 6th prime is 13 and the smallest prime Wieferich base for 13 is 19. Applying this procedure recursively to the resulting bases a total of b-1 = 3 times leads to 911.
Array starts:
  2    5    7   19    127    911     7331   167149    387749  17153317 ...
  3   17  131  659    503   9833    49603   327317  13900147 144229223 ...
  5    7   19  127    911   7331   167149   387749  17153317       ...
  7   19  127  911   7331 167149   387749 17153317 432383657       ...
  11   3   17  131    659    503     9833    49603    327317       ...
  13  19  127  911   7331 167149   387749      ...
  17 131  659  503   9833  49603   327317      ...
  19 127  911 7331 167149 387749 17153317      ...
  23 263   79   31    229    503      ...
  29  41  313 1499    941  12011      ...
  ...

Cf. A001220, A244249.

Second column of table is A125636.

forprime(p=1, 30, b=1; i=0; q=p; print1(p, ", "); while(i < 6, b++; if(Mod(b, q^2)^(q-1)==1 && isprime(b), print1(b, ", "); q=b; b=1; i++)); print(""))

A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

Original entry on oeis.org

2, 5, 8, 7, 7, 17, 18, 15, 26, 7, 3, 17, 19, 19, 26, 31, 38, 53, 28, 7, 19, 3, 28, 17, 57, 19, 80, 19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, 134, 19, 338, 161, 3, 31, 28, 41, 53, 107, 264, 115, 19, 127, 99, 161, 143, 65, 28, 99, 11, 55

Offset: 1

Eric Chen, Feb 21 2015

nonn

a(n) = least base b > 1 such that n is a Wieferich number (see A077816).
At least, b = n^2+1 can satisfy this equation, so a(n) is defined for all n.
Least Wieferich number (>1) to base n: 2, 1093, 11, 1093, 2, 66161, 4, 3, 2, 3, 71, 2693, 2, 29, 4, 1093, 2, 5, 3, 281, 2, 13, 4, 5, 2, ...; each is a prime or 4. It is 4 if and only if n mod 72 is in the set {7, 15, 23, 31, 39, 47, 63}.
Does every natural number (>1) appear in this sequence? If yes, do they appear infinitely many times?
For prime n, a(n) = A185103(n), does there exist any composite n such that a(n) = A185103(n)?

			a(30) = 107 since A000010(30) = 8, 30^2 = 900, and 107 is the least base b > 1 such that b^8 = 1 (mod 900).

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A039678, A185103, A125636, A039951, A247154, A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A077816, A242958, A242959, A241978, A242960, A241977, A253016, A245529.

f[n_] := Block[{b = 2, m = EulerPhi[n]}, While[ PowerMod[b, m, n^2] != 1, b++]; b]; f[1] = 2; Array[f, 72] (* Robert G. Wilson v, Feb 28 2015 *)

a(n)=for(k=2,2^24,if((k^eulerphi(n))%(n^2)==1, return(k)))

a(prime(n)) = A039678(n) = A185103(prime(n)).
a(A077816(n)) = 2.
a(A242958(n)) <= 3.

A289379 Primes p that set a new record for the size of the smallest prime q such that q^(p-1) == 1 (mod p^2), i.e., such that p is a base-q Wieferich prime.

Original entry on oeis.org

2, 3, 7, 17, 23, 37, 67, 89, 139, 163, 269, 379, 439, 491, 691, 701, 877, 1009, 1327, 1427, 1669, 2687, 4973, 6367, 7603, 9277, 10531, 11047, 12071, 18313, 29389, 34471, 42703, 42961, 57731, 77773, 87299, 105517, 113957, 118369, 151303, 192631, 205603, 232091

Offset: 1

Felix Fröhlich, Sep 02 2017

nonn

For n > 1, primes p such that A125636(i) reaches record values, where i is the index of p in A000040.

Cf. A000040, A125636, A287147.

minprimb(n) = forprime(q=1, , if(Mod(q, n^2)^(n-1)==1, return(q)))
my(r=0); forprime(p=1, , if(minprimb(p) > r, print1(p, ", "); r=minprimb(p)))

a(37)-a(44) from Giovanni Resta, Sep 02 2017

A355658 Smallest prime base q such that q^(p-1) == 1 (mod p^2), where p = prime(n).

Original entry on oeis.org

5, 17, 7, 19, 3, 19, 131, 127, 263, 41, 229, 691, 313, 19, 53, 521, 53, 601, 1301, 11, 619, 31, 269, 3187, 53, 181, 43, 317, 499, 373, 911, 659, 19, 3659, 313, 751, 233, 4373, 3307, 419, 2591, 313, 1249, 2897, 349, 709, 331, 1973, 1933, 503, 821, 977, 2371, 263

Offset: 1

Felix Fröhlich, Jul 12 2022

nonn

a(n) differs from A125636(n) if and only if p is a Wieferich prime (A001220). In particular, a(183) = 2 and A125636(183) = 18979. Similarly, a(490) = 2 and A125636(490) = 82183.

Cf. A001220, A125636, A039678.

a(n) = my(p=prime(n)); forprime(q=1, , if(Mod(q, p^2)^(p-1)==1, return(q)))

cp's OEIS Frontend

A133862 Smallest odd prime base q such that p^12 divides q^(p-1) - 1, where p = prime(n).

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A133863 Smallest odd prime base q such that p^13 divides q^(p-1) - 1, where p = prime(n).

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A133864 Smallest odd prime base q such that p^14 divides q^(p-1) - 1, where p = prime(n).

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A133865 Smallest odd prime base q such that p^15 divides q^(p-1) - 1, where p = prime(n).

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A249162 Square array A(p, b) read by antidiagonals in which rows are indexed by successive prime numbers p_i and row b(p_i) gives the smallest prime base b_n to which q = (p_i, b_(n-1)) is a Wieferich prime.

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A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

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A289379 Primes p that set a new record for the size of the smallest prime q such that q^(p-1) == 1 (mod p^2), i.e., such that p is a base-q Wieferich prime.

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A355658 Smallest prime base q such that q^(p-1) == 1 (mod p^2), where p = prime(n).

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