A125635 -id:A125635 - cp's OEIS frontend

A125611 a(n) is the smallest prime p such that 7^n divides p^6 - 1.

2, 19, 19, 3449, 32261, 152617, 3294173, 3376853, 135967277, 135967277, 7909306973, 92233439147, 115385868869, 1356446145697, 56020344873707, 56020344873707, 930522055948829, 9116268492336169, 10744682090246617

Offset: 1

Alexander Adamchuk, Nov 28 2006

nonn

a(n) is the smallest 6th root of unity (mod 7^n) that is prime. - Robert Israel, Jan 14 2024

Robert Israel, Table of n, a(n) for n = 1..1000
Wilfrid Keller and Jörg Richstein, Fermat quotients that are divisible by p, 2014. [Wayback Machine link]

Cf. A125609, A125610, A125612, A125632, A125633, A125634, A125635.

f:= proc(n) local R, r, i;
  R:= sort(map(rhs@op, [msolve(x^6=1, 7^n)]));
  for i from 0 do
    for r in R do
      if isprime(7^n * i + r) then return 7^n * i + r fi
  od od;
end proc:
map(f, [$1..30]); # Robert Israel, Jan 14 2024

PARI
```
\\ See A125609
```

from itertools import count
from sympy import nthroot_mod, isprime
def A125611(n):
    m = 7**n
    r = sorted(nthroot_mod(1,6,m,all_roots=True))
    for i in count(0,m):
        for p in r:
            if isprime(i+p): return i+p # Chai Wah Wu, May 02 2024

More terms from Ryan Propper, Jan 03 2007

More terms from Martin Fuller, Jan 11 2007

A125609 Smallest prime p such that 3^n divides p^2 - 1.

Original entry on oeis.org

2, 17, 53, 163, 487, 1459, 4373, 13121, 39367, 472391, 1062881, 1062881, 19131877, 19131877, 57395627, 86093443, 258280327, 3874204891, 6973568801, 6973568801, 188286357653, 188286357653, 188286357653, 4518872583697, 15251194969973

Offset: 1

Alexander Adamchuk, Nov 28 2006

nonn

Smallest prime of the form k*3^n-1 or k*3^n+1. - Robert Israel, Oct 27 2019

Robert Israel, Table of n, a(n) for n = 1..2088
Martin Fuller, PARI program
W. Keller and J. Richstein, Fermat quotients that are divisible by p.

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. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

f:= proc(n) local k;
      for k from 1 do
        if isprime(k*3^n-1) then return k*3^n-1
        elif isprime(k*3^n+1) then return k*3^n+1
        fi
      od
end proc:
map(f, [$1..30]); # Robert Israel, Oct 27 2019

f[n_] := Module[{k}, For[k = 1, True, k++, If[PrimeQ[k*3^n-1], Return[k*3^n-1], If[PrimeQ[k*3^n+1], Return[k*3^n+1]]]]];
Array[f, 30] (* Jean-François Alcover, Jun 04 2020, after Maple *)

PARI
```
\\ See link.
```

Corrected and extended by Ryan Propper, Jan 01 2007

More terms from Martin Fuller, Jan 11 2007

A125610 Smallest prime p such that 5^n divides p^4 - 1.

Original entry on oeis.org

2, 7, 193, 443, 14557, 14557, 735443, 3124999, 7812499, 78124999, 292968749, 853235443, 2441406251, 53834264557, 122070312499, 202513391693, 1118040735443, 3459595983307, 3459595983307, 270488404577057

Offset: 1

Alexander Adamchuk, Nov 28 2006

Robert Israel, Table of n, a(n) for n = 1..1424
W. Keller and J. Richstein Fermat quotients that are divisible by p.

Cf. A125609 = Smallest prime p such that 3^n divides p^2 - 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. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

f:= proc(n) local k, p2,P,t;
    p2:= numtheory:-msqrt(-1,5^n);
    P:= sort([1,p2,5^n-p2,5^n-1]);
    for k from 0 do
      for t in P do
        if isprime(k*5^n+t) then return k*5^n+t fi
    od od:
end proc:
map(f, [$1..30]); # Robert Israel, Oct 27 2019

\\ See A125609 - Martin Fuller, Jan 11 2007

More terms from Ryan Propper, Jan 02 2007

More terms from Martin Fuller, Jan 11 2007

A125612 a(n) is the smallest prime p such that 11^n divides p^10 - 1.

Original entry on oeis.org

2, 3, 2663, 45989, 275393, 2120879, 28723679, 174625993, 4715895383, 24262286441, 1194631280321, 3143820659087, 138090848575723, 488581592070877, 6266190914259137, 367597838908577287, 10866698414795559631, 19697814061539637951, 19697814061539637951, 3824465353837845574717, 14852046860008834240157

Offset: 1

Alexander Adamchuk, Nov 28 2006

nonn

a(n) is the smallest 10th root of unity (mod 11^n) that is prime. - Robert Israel, Jan 14 2024

Robert Israel, Table of n, a(n) for n = 1..954
W. Keller and J. Richstein, Fermat quotients that are divisible by p.

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. 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. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

f:= proc(n) local R,r,i;
  R:= sort(map(rhs@op, [msolve(x^10=1, 11^n)]));
  for i from 0 do
    for r in R do
      if isprime(11^n * i + r) then return 11^n * i + r fi
  od od;
end proc:
map(f, [$1..20]); # Robert Israel, Jan 14 2024

spp[n_]:=Module[{p=2,c=11^n},While[PowerMod[p,10,c]!=1,p=NextPrime[p]];p]; Array[spp,16] (* Harvey P. Dale, Aug 08 2019 *)

PARI
```
\\ See A125609
```

from itertools import count
from sympy import nthroot_mod, isprime
def A125612(n):
    m = 11**n
    r = sorted(nthroot_mod(1,10,m,all_roots=True))
    for i in count(0,m):
        for p in r:
            if isprime(i+p): return i+p # Chai Wah Wu, May 02 2024

More terms from Ryan Propper, Jan 03 2007
More terms from Martin Fuller, Jan 11 2007
More terms from Robert Israel, Jan 14 2024

A125632 Smallest prime p such that 13^n divides p^12 - 1.

Original entry on oeis.org

2, 19, 239, 239, 220861, 7654109, 533810141, 533810141, 822557039, 38050596989, 2395794301259, 58713568184837, 358661570404751, 22771419458231473, 65106791321062951, 1951482088631313647, 13942901952235522979

Offset: 1

Alexander Adamchuk, Nov 28 2006

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

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. 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. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

PARI
```
See A125609
```

More terms from Martin Fuller, Jan 11 2007

A125633 Smallest prime p such that 17^n divides p^16 - 1.

Original entry on oeis.org

2, 131, 653, 15541, 15541, 24527681, 38277341, 16048035481, 48718117843, 5498076927457, 38413406256881, 2359162908109223, 44510586506850631, 346100334752156863, 12132548193910221893, 201533461539194779193

Offset: 1

Alexander Adamchuk, Nov 28 2006

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

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. A125634 = Smallest prime p such that 19^n divides p^18 - 1. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

\\ See A125609 - Martin Fuller, Jan 11 2007

More terms from Martin Fuller, Jan 11 2007

A125634 Smallest prime p such that 19^n divides p^18 - 1.

Original entry on oeis.org

2, 127, 2819, 2819, 2342959, 2342959, 47579927, 3620189879, 513127081109, 8388044818849, 77460384757423, 2649283656602003, 252317900773542353, 2467410166021233673, 50407811312994280933, 179869204428830533411

Offset: 1

Alexander Adamchuk, Nov 28 2006

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

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. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

\\ See A125609 - Martin Fuller, Jan 11 2007

More terms from Martin Fuller, Jan 11 2007

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

cp's OEIS Frontend

A125611 a(n) is the smallest prime p such that 7^n divides p^6 - 1.

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Maple

PARI

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A125609 Smallest prime p such that 3^n divides p^2 - 1.

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Maple

Mathematica

PARI

Extensions

A125610 Smallest prime p such that 5^n divides p^4 - 1.

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Maple

PARI

Extensions

A125612 a(n) is the smallest prime p such that 11^n divides p^10 - 1.

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Author

Keywords

Comments

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Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions

A125632 Smallest prime p such that 13^n divides p^12 - 1.

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Programs

PARI

Extensions

A125633 Smallest prime p such that 17^n divides p^16 - 1.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A125634 Smallest prime p such that 19^n divides p^18 - 1.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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

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