A098640 -id:A098640 - cp's OEIS frontend

A241573 2^p + 3 where p is prime.

7, 11, 35, 131, 2051, 8195, 131075, 524291, 8388611, 536870915, 2147483651, 137438953475, 2199023255555, 8796093022211, 140737488355331, 9007199254740995, 576460752303423491, 2305843009213693955, 147573952589676412931, 2361183241434822606851

Offset: 1

Vincenzo Librandi, Apr 29 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. sequences of the form 2^p+k with p prime: A034785 (k=0), A001348 (k=-1), A098640 (k=1), A241676 (k=-3), this sequence (k=3), A241678 (k=-5), A241677 (k=5), A098815 (k=-7), A241679 (k=7), A098231 (k=-11), A241680 (k=11).

Magma
```
[2^p+3: p in PrimesUpTo(100)];
```
Mathematica
```
Table[2^Prime[n] + 3, {n, 20}]
```

[2^p+3 for p in primes(100)] # Bruno Berselli, Apr 29 2014

A235712 Least prime p < prime(n) with 2^p + 1 a quadratic nonresidue modulo prime(n), or 0 if such a prime p does not exist.

Original entry on oeis.org

0, 2, 0, 2, 7, 2, 2, 5, 2, 11, 11, 2, 7, 2, 2, 2, 5, 5, 2, 5, 2, 5, 2, 5, 2, 7, 2, 2, 5, 2, 2, 13, 2, 5, 13, 5, 2, 2, 2, 2, 5, 11, 5, 2, 2, 7, 5, 2, 2, 23, 2, 7, 5, 5, 2, 2, 5, 5, 2, 7, 2, 2, 2, 5, 2, 2, 7, 2, 2, 5, 2, 7, 2, 2, 11, 2, 5, 2, 5, 5, 5, 7, 7, 2, 5, 2, 5, 2, 7, 2, 2, 7, 2, 13, 7, 2, 5, 5, 2, 5

Offset: 1

Zhi-Wei Sun, Apr 20 2014

nonn

Conjecture: a(n) > 0 for all n > 3.
Note that 2^3 + 1 = 3^2 is a quadratic residue modulo any prime p > 3. Also, there is no prime p < prime(316) = 2089 with 2^p + 1 a primitive root modulo 2089.
See also A234972 and A235709 for similar conjectures.

			a(4) = 2 since 2^2 + 1 = 5 is a quadratic nonresidue modulo prime(4) = 7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A098640, A234972, A235709.

Do[Do[If[JacobiSymbol[2^(Prime[k])+1,Prime[n]]==-1,Print[n," ",Prime[k]];Goto[aa]],{k,1,n-1}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

A152099 a(n) = (2^prime(n) - 1)(2^prime(n) + 1) = 2^(2prime(n)) - 1.

Original entry on oeis.org

15, 63, 1023, 16383, 4194303, 67108863, 17179869183, 274877906943, 70368744177663, 288230376151711743, 4611686018427387903, 18889465931478580854783, 4835703278458516698824703, 77371252455336267181195263, 19807040628566084398385987583

Offset: 1

Roger L. Bagula, Nov 24 2008

nonn

Idea resulted from seqfan posts by Artur Jasinski.

Cf. A000040, A000302, A001348, A034785, A098640.

Table[(2^Prime[n] - 1)*(2^Prime[n] + 1), {n, 1, 20}]

from sympy import prime
def A152099(n): return (1<<(prime(n)<<1))-1 # Chai Wah Wu, Jun 26 2023

a(n) = A001348(n) * A098640(n).
a(n) = A034785(n)^2 - 1.
a(n) = A000302(A000040(n)) - 1.

A184084 Decimal expansion of product_{p=primes} (1-1/(2^p+1)).

Original entry on oeis.org

6, 8, 3, 7, 9, 2, 8, 4, 2, 3, 5, 9, 4, 7, 4, 3, 6, 8, 9, 9, 4, 3, 6, 3, 6, 4, 3, 4, 1, 0, 7, 6, 3, 4, 4, 4, 3, 6, 8, 2, 2, 1, 0, 8, 7, 5, 8, 1, 8, 1, 7, 3, 5, 2, 6, 2, 9, 4, 7, 1, 2, 9, 8, 1, 0, 5, 5, 9, 9, 0, 4, 2, 3, 5, 5, 6, 5, 5, 7, 7, 7

Offset: 0

R. J. Mathar, Jan 09 2011

Inverse of the constant A184083.

			(1-1/5) *(1-1/9) *(1-1/33) * (1-1/129) *(1-1/2049)* ... = 0.6837928423594743689943636...

RealDigits[Times@@Table[1-1/(2^p+1),{p,Prime[Range[1000]]}],10,100][[1]] (* Harvey P. Dale, Jul 23 2024 *)

Equals product_{p in A000040} (1-1/(2^p+1)) = 1/ product_p (1+2^(-p)) = product_{n>=1} (1-1/A098640(n)).

A260674 Primes p for which the greatest common divisor of 2^p+1 and 3^p+1 is greater than 1.

Original entry on oeis.org

2, 83, 107, 367, 569, 887, 1327, 1451, 1621, 1987, 2027, 3307, 3547, 3631, 3691, 4421, 4547, 4967, 5669, 5843, 5927, 6011, 6911, 6991, 7207, 7949, 8167, 8431, 10771, 10889, 11287, 11621, 12007, 12227, 12487, 12763, 12983, 15391, 15767, 16127, 17107, 17183, 17231

Offset: 1

Alex Jordan, Nov 14 2015

nonn

Primes p such that A066803(p)>1. - Tom Edgar, Nov 15 2015

			Since GCD(2^83 + 1, 3^83 + 1) = 499, the prime 83 is in the sequence. It is only the second such prime, so a(2) = 83.

Chai Wah Wu, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 1064. GCD(2^p+1,3^p+1), The Prime Puzzles and Problems Connection.

Cf. A066803, A098640, A349722.

Select[Prime@ Range@ 2000, GCD[2^# + 1, 3^# + 1] > 1 &] (* Michael De Vlieger, Nov 16 2015 *)

list(lim)=forprime(p=2,lim,if(gcd(2^p+1,3^p+1)>1,print1(p, ", "))) \\ Anders Hellström, Nov 14 2015

from sympy import prime
from math import gcd
A260674_list = [p for p in (prime(n) for n in range(1,10**3)) if gcd(2**p+1,3**p+1) > 1] # Chai Wah Wu, Nov 23 2015

# code will list all such primes no larger than the N-th prime.
N=1000
for k in range(N):
    if (gcd(2^Primes().unrank(k)+1,3^Primes().unrank(k)+1) != 1):
        print(Primes().unrank(k))

A260507 Primes p such that (2^p+1)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

2, 7, 179, 619, 17807

Offset: 1

Felix Fröhlich, Jul 27 2015

A000040(n) such that A260531(n) = 1.
Is this a subsequence of A130060?
a(6) > 10325801 if it exists.
a(6) > 3037000499 if it exists. - Hiroaki Yamanouchi, Aug 20 2015

			2^7 + 1 = 129 and 129^6 == 1 (mod 7^2), so 7 is a term of the sequence.

Cf. A098640, A127074, A130060, A130062, A260531.

Select[Prime@ Range@ 120, Mod[(2^# + 1)^(# - 1), #^2] == 1 &] (* Michael De Vlieger, Jul 29 2015 *)

forprime(p=2, , if(Mod(2^p+1, p^2)^(p-1)==1, print1(p, ", ")))

A260531 a(n) = (2^p+1)^(p-1) modulo p^2, where p is prime(n).

Original entry on oeis.org

1, 0, 21, 1, 45, 79, 120, 305, 484, 697, 404, 186, 1354, 603, 612, 2757, 945, 3051, 3552, 498, 950, 1186, 2657, 1781, 6403, 9192, 8035, 1927, 2181, 2713, 6097, 2621, 10139, 3476, 10878, 8608, 22609, 21028, 24550, 19031, 1, 12852, 33426, 27793, 34279, 11543

Offset: 1

Felix Fröhlich, Jul 28 2015

nonn

The primes where a(n) == 1 are given by A260507.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A000040, A098640, A260507.

f[n_] := Block[{p = Prime@ n}, PowerMod[2^p + 1, p - 1, p^2]]; Array[f, 46] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = lift(Mod(2^prime(n)+1, prime(n)^2)^(prime(n)-1))

a(n) = A098640(n)^(A000040(n)-1) modulo A000040(n)^2.

A174623 (2^p+1)^2 where p is prime.

Original entry on oeis.org

25, 81, 1089, 16641, 4198401, 67125249, 17180131329, 274878955521, 70368760954881, 288230377225453569, 4611686022722355201, 18889465931753458761729, 4835703278462914745335809, 77371252455353859367239681, 19807040628566365873362698241

Offset: 1

Vincenzo Librandi, Apr 11 2010

nonn

			For p=2, (2^2+1)^2=25;
for p=3, (2^3+1)^2=81;
for p=5, (2^5+1)^2=1089.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Magma
```
[(2^p+1)^2: p in [0..1000] |IsPrime(p)]
```

A174623:=n->(2^ithprime(n)+1)^2: seq(A174623(n), n=1..20); # Wesley Ivan Hurt, Apr 18 2017

Table[(2^Prime[n] + 1)^2, {n, 30}] (* Vincenzo Librandi, Apr 29 2014 *)

a(n) = (A098640(n))^2. - R. J. Mathar, Jul 06 2010

Definition clarified by Jon E. Schoenfield, Jun 18 2010

A247939 Sum of divisors of 2^prime(n)+1.

Original entry on oeis.org

6, 13, 48, 176, 2736, 10928, 174768, 699056, 11184816, 727960800, 2863311536, 183355069408, 2967356682528, 11728124029616, 188313058624992, 12121838249371488, 768906329487027264, 3074457345618258608, 196765296972010592800, 3148244377723715041632

Offset: 1

Vincenzo Librandi, Sep 27 2014

nonn

b-file computed with factorizations in Wagstaff link. a(166) corresponding to 2^983+1 is currently the first unknown term. - Jens Kruse Andersen, Sep 28 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..165
Sam Wagstaff, The Cunningham Project

Cf. A000203, A069061, A098640.

[SumOfDivisors(2^p+1): p in PrimesUpTo(100)];

with(numtheory): A247939:=n->sigma(2^ithprime(n)+1): seq(A247939(n), n=1..20); # Wesley Ivan Hurt, Sep 27 2014

Table[DivisorSigma[1, 2^Prime[n] + 1], {n, 1, 20}]

vector(50,n,sigma(2^prime(n)+1)) \\ Derek Orr, Sep 27 2014

a(n) = A000203(A098640(n)). - Michel Marcus, Sep 27 2014

A098773 p*2^p + 1 where p is prime.

Original entry on oeis.org

9, 25, 161, 897, 22529, 106497, 2228225, 9961473, 192937985, 15569256449, 66571993089, 5085241278465, 90159953477633, 378231999954945, 6614661952700417, 477381560501272577, 34011184385901985793

Offset: 1

Parthasarathy Nambi, Oct 30 2004

nonn

			If p=3, then 3*2^3 + 1 = 25.

Cf. A098640.

Table[p = Prime[n]; p*2^p + 1, {n, 17}] (* Robert G. Wilson v, Nov 04 2004 *)

More terms from Robert G. Wilson v, Nov 04 2004

cp's OEIS Frontend

A241573 2^p + 3 where p is prime.

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A235712 Least prime p < prime(n) with 2^p + 1 a quadratic nonresidue modulo prime(n), or 0 if such a prime p does not exist.

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A152099 a(n) = (2^prime(n) - 1)*(2^prime(n) + 1) = 2^(2*prime(n)) - 1.

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A184084 Decimal expansion of product_{p=primes} (1-1/(2^p+1)).

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A260674 Primes p for which the greatest common divisor of 2^p+1 and 3^p+1 is greater than 1.

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A260507 Primes p such that (2^p+1)^(p-1) == 1 (mod p^2).

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A260531 a(n) = (2^p+1)^(p-1) modulo p^2, where p is prime(n).

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A174623 (2^p+1)^2 where p is prime.

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A152099 a(n) = (2^prime(n) - 1)(2^prime(n) + 1) = 2^(2prime(n)) - 1.