A125611 -id:A125611 - cp's OEIS frontend

A369154 Numbers k such that A125611(k) = A125611(k + 1).

2, 9, 15, 28, 40, 41, 42, 48, 60, 68, 79, 83, 93, 95, 98, 100, 108, 114, 118, 120, 124, 129, 132, 137, 147, 149, 167, 196, 202, 206, 207, 215, 219, 221, 223, 225, 230, 243, 248, 255, 261, 265, 274, 276, 287, 299, 302, 320, 323, 329, 337, 341, 353, 356, 360, 364, 365, 373, 380, 381, 391, 405, 410

Offset: 1

Robert Israel, Jan 14 2024

nonn

Numbers k such that A125611(k)^6 - 1 is divisible by 7^(k+1).

Since the 3 consecutive numbers 40, 41 and 42 are in the sequence, A125611(40) = A125611(41) = A125611(42) = A125611(43).

			a(3) = 15 is a term because A125611(15) = A125611(16) = 56020344873707, i.e., 56020344873707 is the least prime p such that p^6 - 1 is divisible by 7^15, and in this case p^6 - 1 is also divisible by 7^16.

Robert Israel, Table of n, a(n) for n = 1..430

Cf. A125611.

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:
R:= NULL: count:= 0:
for k from 1 while count < 100 do
 v:= f(k);
 if v = w then R:= R, k-1; count:= count+1 fi;
 w:= v;
od:
R;

from itertools import count, islice
from sympy import nthroot_mod, isprime
def A369154_gen(): # generator of terms
    c, m = 1, 1
    for k in count(0):
        m *= 7
        r = sorted(nthroot_mod(1,6,m,all_roots=True))
        for i in count(0,m):
            for p in r:
                if isprime(i+p):
                    if i+p == c:
                        yield k
                    c = i+p
                    break
            else:
                continue
            break
A369154_list = list(islice(A369154_gen(),30)) # Chai Wah Wu, May 04 2024

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

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

Original entry on oeis.org

2, 1993, 134857, 716192579, 68539500613, 101479854517477, 711236716682257, 1646895113182602793, 783453821802171722617, 91545091731109499684503, 5225628509593228805996529, 1808125915932987167909775139

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

PARI
```
See A125609
```

More terms from Martin Fuller, Jan 11 2007

A125636 Smallest odd prime base q such that p^2 divides q^(p-1) - 1, 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

Alexander Adamchuk, Nov 28 2006

nonn

Robert Israel, Table of n, a(n) for n = 1..1000
W. Keller and J. Richstein, Fermat quotients that are divisible by p.

Cf. A125637 (analogous with p^3 instead of p^2).

Cf. A125609 (q=3), A125610 (q=5), A125611 (q=7), A125612 (q=11), A125632 (q=13), A125633 (q=17), A125634 (q=19): sequences of smallest prime p such that q^n divides p^(q-1) - 1.

a:= proc(p)
  local q;
  q:= 3;
  while (q &^ (p-1) - 1) mod p^2 <> 0 do
    q:= nextprime(q)
  od:
  q
end proc:
seq(a(ithprime(n)), n=1..100); # Robert Israel, Nov 24 2014

Table[Function[p, q = 3; While[! Divisible[q^(p - 1) - 1, p^2], q = NextPrime@ q]; q]@ Prime@ n, {n, 54}] (* Michael De Vlieger, Feb 12 2017 *)

a(n) = {p = prime(n); forprime(q=3, , if (Mod(q, p^2)^(p-1) == 1, return (q)););} \\ Michel Marcus, Nov 24 2014

Removed an incorrect comment. - Felix Fröhlich, Feb 12 2017

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A369154 Numbers k such that A125611(k) = A125611(k + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs