A125636 -id:A125636 - cp's OEIS frontend

A125637 Smallest odd prime base q such that p^3 divides q^(p-1) - 1, where p = prime(n).

17, 53, 193, 19, 2663, 239, 653, 2819, 13931, 10133, 6287, 691, 10399, 3623, 6397, 9283, 63463, 38447, 36809, 21499, 75227, 1523, 55933, 42937, 341293, 4943, 255007, 5573, 56633, 262079, 94961, 33289, 65543, 298157, 218579, 25667, 411589, 253987

Offset: 1

Alexander Adamchuk, Nov 28 2006

nonn

W. Keller and J. Richstein Fermat quotients that are divisible by p.

Cf. A125636 = Smallest odd prime base q such that p^2 divides q^(p-1) - 1, where p = Prime[n]. Cf. A125609 = Smallest prime p such that 3^n divides p^2 - 1. Cf. A125610 = Smallest prime p such that 5^n divides p^4 - 1. Cf. A125611 = Smallest prime p such that 7^n divides p^6 - 1. Cf. A125612 = Smallest prime p such that 11^n divides p^10 - 1. Cf. A125632 = Smallest prime p such that 13^n divides p^12 - 1. Cf. A125633 = Smallest prime p such that 17^n divides p^16 - 1. Cf. A125634 = Smallest prime p such that 19^n divides p^18 - 1.

A125645 Smallest odd prime base q such that p^4 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

17, 163, 443, 3449, 45989, 239, 15541, 2819, 60793, 78017, 690143, 398023, 1977343, 574081, 1513367, 4388179, 3198427, 8065789, 3246107, 1353383, 5934307, 15631613, 2864371, 14754769, 15012733, 1358891, 32414783, 119551, 21860063, 11281097

Offset: 1

Alexander Adamchuk, Nov 29 2006

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
W. Keller and J. Richstein Fermat quotients that are divisible by p.

Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125646, A125647, A125648, A125649.

f:= proc(n) local p, r, S,i,s,t;
  uses numtheory;
  p:= ithprime(n);
  r:= primroot(p^4);
  S:= sort([seq(r &^ (i*p^3) mod p^4, i=0..p-2)]);
  for i from 0 do
    for s in S do
      t:= i*p^4+s;
      if t::odd and isprime(t) then return t fi
  od od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Feb 12 2017

{ a(n) = local(p,x,y); if(n==1,return(17)); p=prime(n); x=znprimroot(p^4)^(p^3); vecsort( vector(p-1,i, y=lift(x^i);while(!isprime(y),y+=p^4);y ) )[1] } \\ Max Alekseyev, May 30 2007

More terms from Max Alekseyev, May 30 2007

A125646 Smallest odd prime base q such that p^5 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

97, 487, 14557, 32261, 275393, 220861, 15541, 2342959, 1051847, 24639193, 40373093, 70697317, 31851901, 47289133, 456330179, 10000453, 154075723, 130702609, 304154189, 143584109, 183298237, 79451167, 1058782027, 352845203, 567620413, 4592184511, 5890772963, 9651540247, 4081988041, 4772484029

Offset: 1

Alexander Adamchuk, Nov 29 2006

nonn

Robert Israel, Table of n, a(n) for n = 1..500
W. Keller and J. Richstein, Fermat quotients that are divisible by p.

Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125647, A125648, A125649.

f:= proc(n) local p,k,j,q,R;
  p:= ithprime(n);
  R:= sort(map(rhs@op, [msolve(q^(p-1)-1, p^5)]));
  for k from 0 do
    for j in R do
      q:= k*p^5+j;
      if isprime(q) then return q fi;
    od
 od
end proc:
map(f, [$1..100]); # Robert Israel, Apr 11 2019

Do[p = Prime[n]; q = 2; While[PowerMod[q, p-1, p^5] != 1, q = NextPrime[q]]; Print[q], {n, 100}] (* Ryan Propper, Mar 31 2007 *)

{ a(n) = local(p,x,y); if(n==1,return(97)); p=prime(n); x=znprimroot(p^5)^(p^4); vecsort( vector(p-1,i, y=lift(x^i);while(!isprime(y),y+=p^5);y ) )[1] } \\ Max Alekseyev, May 30 2007

from itertools import count
from sympy import nthroot_mod, isprime, prime
def A125646(n):
    m = (p:=prime(n))**5
    r = sorted(nthroot_mod(1,p-1,m,all_roots=True))
    for i in count(0,m):
        for a in r:
            if isprime(i+a): return i+a # Chai Wah Wu, May 02 2024

More terms from Ryan Propper, Mar 31 2007

More terms from Max Alekseyev, May 30 2007

A125648 Smallest odd prime base q such that p^7 divides q^(p-1) - 1, where p = Prime[n].

Original entry on oeis.org

257, 4373, 735443, 3294173, 28723679, 533810141, 38277341, 47579927, 982740799, 33956348611, 77141582851, 174329354539, 82984891817, 109051450427, 83209719751, 1352085061013, 171168499897, 1822904926391, 2870322429133, 3589197993463, 2603594622571, 5834621843669, 1411025860033, 20635686238253, 1580041060459, 26763849212297, 8216934406781, 28482190726739, 97876187600351

Offset: 1

Alexander Adamchuk, Nov 29 2006

nonn

W. Keller and J. Richstein Fermat quotients that are divisible by p.

Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125649.

{ a125648(n) = my(p, x, r); if(n==1, return(257)); p=prime(n); x=znprimroot(p^7)^(p^6); vecmin( vector(p-1, i, forprimestep(y=2,oo,x^i,r=y;break); r) ); } \\ Max Alekseyev, May 30 2007; updated Apr 01 2025

More terms from Max Alekseyev, May 30 2007

A125647 Smallest odd prime base q such that p^6 divides q^(p-1) - 1, where p = Prime[n].

Original entry on oeis.org

193, 1459, 14557, 152617, 2120879, 7654109, 24527681, 2342959, 90603883, 1657641497, 40373093, 2175429661, 1614357949, 119612113, 14635471219, 2816276179, 15591204869, 1006953931, 7726467079, 48931161299, 54908441659, 41985419521, 583493688221, 200335697059, 96891225583, 50303508131, 129847013561, 362253784469, 625810253147, 195406393583

Offset: 1

Alexander Adamchuk, Nov 29 2006

nonn

W. Keller and J. Richstein Fermat quotients that are divisible by p.

Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125648, A125649.

{ a(n) = local(p,x,y); if(n==1,return(193)); p=prime(n); x=znprimroot(p^6)^(p^5); vecsort( vector(p-1,i, y=lift(x^i);while(!isprime(y),y+=p^6);y ) )[1] } \\ Max Alekseyev, May 30 2007

More terms from Max Alekseyev, May 30 2007

A125649 Smallest odd prime base q such that p^8 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

257, 13121, 3124999, 3376853, 174625993, 533810141, 16048035481, 3620189879, 982740799, 547344139109, 497929938133, 1105109875657, 15682480615619, 1391016035411, 83209719751, 84224951222611, 165554755409789, 254747341131683, 701000310909907, 317304132615017

Offset: 1

Alexander Adamchuk, Nov 29 2006

nonn

W. Keller and J. Richstein Fermat quotients that are divisible by p.

Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648.

Do[p = Prime[n]; q = 2; While[PowerMod[q, p-1, p^8] != 1, q = NextPrime[q]]; Print[q], {n, 100}] (* Ryan Propper, Apr 01 2007 *)

{ a(n) = local(p,x,y); if(n==1,return(257)); p=prime(n); x=znprimroot(p^8)^(p^7); vecsort( vector(p-1,i, y=lift(x^i);while(!isprime(y),y+=p^8);y ) )[1] } \\ Max Alekseyev, May 30 2007

More terms from Ryan Propper, Apr 01 2007

More terms from Max Alekseyev, May 30 2007

A174422 1st Wieferich prime base prime(n).

Original entry on oeis.org

1093, 11, 2, 5, 71, 2, 2, 3, 13, 2, 7, 2, 2, 5

Offset: 1

Jonathan Sondow, Mar 19 2010

Smallest prime p such that p^2 divides prime(n)^(p-1) - 1.
Smallest prime p such that p divides the Fermat quotient q_p((prime(n)) = (prime(n)^(p-1) - 1)/p.
See additional comments, links, and cross-refs in A039951.
a(15) = A039951(47) > 4.1*10^13.

			a(1) = 1093 is the first Wieferich prime A001220. a(2) = 11 is the first Mirimanoff prime A014127.

Wikipedia, Generalized Wieferich primes.
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (or 0 if unknown)

Cf. A001220, A014127, A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A125636 = smallest prime p such that prime(n)^2 divides p^(prime(n)-1) - 1.

Cf. A178871 = 2nd Wieferich prime base prime(n).

f[n_] := Block[{b = Prime@ n, p = 2}, While[ PowerMod[b, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 14]

forprime(a=2, 20, forprime(p=2, 10^9, if(Mod(a, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jun 27 2014

a(n) = A039951(prime(n)).

a(n) = 2 if and only if prime(n) == 1 (mod 4). [Jonathan Sondow, Aug 29 2010]

A133859 Smallest odd prime base q such that p^9 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

7681, 39367, 7812499, 135967277, 4715895383, 822557039, 48718117843, 513127081109, 147534349327, 21203414421907, 52879244321341, 15069267560119, 798099274499279, 164129642266943, 1740228634955257, 149381307185023

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(9) = 7681.
a(2) = A125609(9) = 39367.
a(3) = A125610(9) = 7812499.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133860, A133861, A133862, A133863, A133864, A133865.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^9 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

A133860 Smallest odd prime base q such that p^10 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

12289, 472391, 78124999, 135967277, 24262286441, 38050596989, 5498076927457, 8388044818849, 30794280412669, 45941644105613, 1205285836084793, 7909086479714171, 1438991183761177, 47101607991825047, 18067554193458689

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(10) = 12289.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133859, A133861, A133862, A133863, A133864, A133865.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^10 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

A133861 Smallest odd prime base q such that p^11 divides q^(p-1) - 1, where p = prime(n).

Original entry on oeis.org

12289, 1062881, 292968749, 7909306973, 1194631280321, 2395794301259, 38413406256881, 77460384757423, 30794280412669, 4161130688896397, 3748333074529501, 240404931594746129, 191828075390557213

Offset: 1

Alexander Adamchuk, Sep 26 2007

nonn

			a(1) = A035089(11) = 12289.

Cf. A035089, A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125646, A125647, A125648, A125649, A133859, A133860, A133862, A133863, A133864, A133865.

Do[ k = 1; While[ !PowerMod[ Prime[ k ], Prime[ n ] - 1, Prime[ n ]^11 ] == 1, k++ ]; Print[ { n, Prime[ k ] } ], {n, 1, 100} ]

Extended by Max Alekseyev, May 08 2009

cp's OEIS Frontend

A125637 Smallest odd prime base q such that p^3 divides q^(p-1) - 1, where p = prime(n).

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A125645 Smallest odd prime base q such that p^4 divides q^(p-1) - 1, where p = prime(n).

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A125646 Smallest odd prime base q such that p^5 divides q^(p-1) - 1, where p = prime(n).

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A125648 Smallest odd prime base q such that p^7 divides q^(p-1) - 1, where p = Prime[n].

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A125647 Smallest odd prime base q such that p^6 divides q^(p-1) - 1, where p = Prime[n].

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A125649 Smallest odd prime base q such that p^8 divides q^(p-1) - 1, where p = prime(n).

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Mathematica

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A174422 1st Wieferich prime base prime(n).

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A133859 Smallest odd prime base q such that p^9 divides q^(p-1) - 1, where p = prime(n).

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A133860 Smallest odd prime base q such that p^10 divides q^(p-1) - 1, where p = prime(n).

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