A035089 -id:A035089 - cp's OEIS frontend

A261117 Smallest positive integer b such that b^(2^n)+1 is divisible by the square of A035089(n+1).

8, 7, 110, 40, 1497, 894, 315, 48, 166107, 95853, 63609, 71589, 492348, 209628, 388440, 48853, 6118793, 2684186, 25787045, 49643800, 54302036, 3969770538, 17592956651, 7347360617, 991255542, 8249087392, 11518171450, 51385581002, 2268777293, 21252616802, 2822082710511

Offset: 0

Jeppe Stig Nielsen, Aug 08 2015

nonn

For given n, if A035089(n+1) exists (which is true by Dirichlet's theorem on arithmetic progressions), then a(n) exists. Proof: p := A035089(n+1) is a prime of the form p=k*2^(n+1)+1, then the group (Z/(p^2)Z)* is cyclic of order p*(p-1) = p*k*2^(n+1). It therefore has an element b of order exactly 2^(n+1). For that b we have then b^(2^n) == -1 (mod p^2).

For given n, a(n) is not necessarily the smallest b such that b^(2^n)+1 is nonsquarefree; see A260824.

			Consider n=4, hence generalized Fermat numbers b^16+1. The first prime (A035089(4+1)) of the form 32*k+1 is 97. It follows that 97 is the smallest prime whose square divides a number of the form b^16+1. The first time 97^2 divides b^16+1 is for b=1497. Hence a(4)=1497. However, A260824(4) is smaller, A260824(4)=392. This is because already 392^16+1 is nonsquarefree (but the prime with a square dividing it, 769, exceeds 97).

Cf. A260824, A248214, A035089.

a(n)=for(k=1,10^10,p=(k<<(n+1))+1;if(isprime(p),break()));for(b=1,p^2,b%p!=0&Mod(b,p^2)^(1<

a(n)=for(k=1, 10^10, p=(k<<(n+1))+1; if(isprime(p), break())); e=p*(p-1)/(1<<(n+1)); h=znprimroot(p^2)^e; g=h^2; m=p^2; for(i=1,1<

A057775 a(n) is the least prime p such that p-1 is divisible by 2^n and not by 2^(n+1).

Original entry on oeis.org

2, 3, 5, 41, 17, 97, 193, 641, 257, 7681, 13313, 18433, 12289, 40961, 114689, 163841, 65537, 1179649, 786433, 5767169, 7340033, 23068673, 104857601, 377487361, 754974721, 167772161, 469762049, 2013265921, 3489660929, 12348030977, 3221225473, 75161927681

Offset: 0

Labos Elemer, Nov 02 2000

nonn

If we drop the requirement that p-1 must not be divisible by 2^(n+1), we get instead A035089, which is a nondecreasing sequence. - Jeppe Stig Nielsen, Aug 09 2015

			a(13) = 40961 = 1 + 8192*5 where the last term is divisible by the 13th power of 2 and 40961 is the smallest prime with that property.

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A000040, A006093, A035050, A035089, A057776, A126717, A201914.

f:= proc(n) local p;
  for p from 2^n+1 by 2^(n+1) do
    if isprime(p) then return p fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, Aug 10 2015

Table[k = 1; While[p = k*2^n + 1; ! PrimeQ[p], k = k + 2]; p, {n, 0, 40}] (* T. D. Noe, Dec 27 2011 *)

a(n)=forstep(k=1,9e99,2,isprime((k<Jeppe Stig Nielsen, Aug 09 2015

a(n) = prime(A057776(n+1)). - Amiram Eldar, Mar 16 2025

More terms from Larry Reeves (larryr(AT)acm.org), Nov 03 2000

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

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

Original entry on oeis.org

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

A334296 Smallest k such that (2k+1)*2^n+1 is prime.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 1, 2, 0, 7, 6, 4, 1, 2, 3, 2, 0, 4, 1, 5, 3, 5, 12, 22, 22, 2, 3, 7, 6, 11, 1, 17, 21, 4, 37, 29, 1, 7, 7, 2, 13, 1, 4, 4, 7, 17, 9, 13, 7, 11, 3, 8, 3, 25, 24, 2, 13, 14, 49, 13, 15, 26, 52, 4, 12, 4, 1, 4, 15, 11, 19, 19, 63, 11, 33, 2, 46

Offset: 0

Mike Speciner, Apr 21 2020

nonn

A057775 is the corresponding sequence of primes.

			a(0)=a(1)=a(2)=0 because 2^0+1=2, 2^1+1=3, 2^2+1=5 are prime.
a(3)=2 because 2^8+1=9 and 3*2^8+1=25 are not prime, but 5*2^8+1=41 is.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A057778, A057775, A035050, A035089.

f:= proc(n) local t, v, k;
   t:= 2^n; v:= -t+1;
   for k from 0 do
      v:= v+2*t;
      if isprime(v) then return k fi
   od
end proc:
map(f, [$0..100]); # Robert Israel, Jul 14 2020

a[n_] := Block[{k = 0}, While[! PrimeQ[(2 k + 1) 2^n + 1], k++]; k]; Array[a, 77, 0] (* Giovanni Resta, May 08 2020 *)

a(n) = my(k=0); while (!isprime((2*k+1)*2^n+1), k++); k; \\ Michel Marcus, Apr 30 2020

from itertools import count
from sympy import isprime
def pow2p1() : # generates the sequence
  for n in count() :
    for k in count() :
      if isprime(((2*k+1)<

a(n) = (A057778(n)-1)/2.

a(n) = ((A057775(n)-1)/2^n-1)/2.

cp's OEIS Frontend

A261117 Smallest positive integer b such that b^(2^n)+1 is divisible by the square of A035089(n+1).

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A057775 a(n) is the least prime p such that p-1 is divisible by 2^n and not by 2^(n+1).

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

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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).