A001348 -id:A001348 - cp's OEIS frontend

A235709 Least prime p < prime(n) with 2^p - 1 a quadratic residue modulo prime(n), or 0 if such a number does not exist.

0, 0, 0, 0, 2, 2, 7, 3, 2, 3, 3, 2, 5, 5, 2, 3, 2, 2, 7, 2, 2, 5, 2, 23, 2, 5, 3, 2, 2, 3, 5, 2, 3, 3, 3, 5, 2, 11, 2, 5, 2, 2, 2, 2, 3, 3, 11, 3, 2, 2, 3, 2, 2, 2, 5, 2, 7, 3, 2, 3, 3, 5, 3, 2, 2, 3, 5, 2, 2, 2, 7, 2, 3, 2, 7, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 11, 5, 2, 2, 5, 2, 5, 2, 7, 5, 3, 2

Offset: 1

Zhi-Wei Sun, Apr 20 2014

nonn

Conjecture: a(n) > 0 for all n > 4.
We have verified this for all n = 5, ..., 10^8.
Note that the conjecture in A234972 implies that for any prime p > 3 there is a prime q < p with 2^q - 1 a quadratic nonresidue modulo p.

			a(8) = 3 since 2^3 - 1 = 7 is a quadratic residue modulo prime(8) = 19, but 2^2 - 1 = 3 is not.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001348, A234972.

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}]

A241573 2^p + 3 where p is prime.

Original entry on oeis.org

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

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

Original entry on oeis.org

5, 9, 12, 13, 17, 19, 23, 25, 26, 27, 29, 32, 33, 34, 35, 39, 45, 46, 49, 53, 57, 58, 60, 62, 69, 74, 75, 82, 88, 93, 99, 129, 140, 152, 164, 166, 168, 178, 179

Offset: 1

Artur Jasinski, Dec 09 2007

a(40)>=206. - Amiram Eldar, Sep 29 2018

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, n]]], {n, 1, 40}]; k
Select[Range[40],PrimeNu[2^Prime[#]-1]==2&] (* Harvey P. Dale, Jul 07 2013 *)

Equals {k: A001221(A001348(k)) = 2}. a(n) = A049084(A135978(n)). - R. J. Mathar, May 03 2008

Edited by R. J. Mathar, May 03 2008
a(17)-a(34) from Donovan Johnson, Jun 14 2009
a(35)-a(39) from Amiram Eldar, Sep 29 2018

A188133 Primes p such that 10p+1 divides 2^p-1.

Original entry on oeis.org

43, 487, 547, 571, 883, 1459, 1663, 1723, 2539, 3319, 3511, 4903, 5107, 5431, 6199, 6367, 8011, 8599, 9007, 9391, 9511, 10111, 11119, 11959, 12379, 12703, 13291, 13339, 13999, 14083, 14551, 14767, 15187, 15319, 15859, 15991, 16183, 16603, 16747, 17659, 18427, 19699

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

It is known that divisors of M(p)=2^p-1 are of the form 2kp+1. For k=1, these are the Lucasian primes A002515, for k=2 there are no such divisors, for k=3 these divisors are listed in A188130 and for k=4 in A122095.
The equivalent sequence of prime indices is 14, 93, 101, 105, 153, 232, 261, 269, ....
If k == 2 (mod 4), there are no such divisors in general and here there are no smaller k's than k = 5. - Karl-Heinz Hofmann, Jan 27 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A350702 (k = 7).

Cf. A001348, A000040.

Select[Range[2*10^4], PrimeQ[#] && PowerMod[2, #, 10# + 1] == 1 &] (* Amiram Eldar, Nov 13 2019 *)
Select[Prime[Range[2500]],PowerMod[2,#,10#+1]==1&] (* Harvey P. Dale, Dec 08 2024 *)

forprime(p=1,1e5, Mod(2,p*10+1)^p-1 || print1(p", "))

from sympy import sieve
print([p for p in sieve[1:10000] if pow(2,p,10*p+1) == 1])
# Karl-Heinz Hofmann, Jan 27 2022

{p = A000040(i): 10*p+1 | A001348(i)}. - R. J. Mathar, Mar 21 2011

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Peter Luschny, Mar 03 2018

nonn

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.

An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

Peter Luschny, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Indices of the nonzero values of A300333.

Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

A066538 Sum of the digits of the n-th Mersenne prime (A000668).

Original entry on oeis.org

3, 7, 4, 10, 19, 13, 28, 46, 73, 112, 139, 154, 697, 847, 1675, 3106, 3106, 4258, 5755, 5950, 13216, 13693, 14980, 27202, 28939, 31339, 60337, 116455, 149365, 179488, 291745, 1026544, 1163443, 1704376, 1893388, 4038358, 4092673, 9440671, 18243946, 28445131, 32580433, 35170384, 41201947, 44142151, 50349694, 57766339, 58416637

Offset: 1

Robert G. Wilson v, Jan 06 2002

From Gord Palameta, Jul 21 2018: (Start)
a(38) and a(39) were calculated by Enoch Haga, Sep 07 1999 and Dec 17 2001; a(40) through a(42) were calculated by Andrew Rupinski, Mar 12 2005. (See the Carlos Rivera link.)
It appears that asymptotically a(n)/A000043(n) = 9*log_10(2)/2. (End)

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Gord Palameta)
Carlos Rivera, Puzzle 74.- SOD(2^6972593-1), The Prime Puzzles & Problems Connection.

Cf. A000043, A000668, A007953, A001348.
Subsequence of: A007953, A007605.
Cf. A001370 (sum of digits of 2^n).

ep = {the exponents from A000043}; a = {}; Do[ a = Append[a, Apply[ Plus, IntegerDigits[ 2^ep[[n]] - 1]]], {n, 1, 47} ]; a
(* Second program: *)
Array[Total@ IntegerDigits[2^MersennePrimeExponent@ # - 1] &, 45] (* Michael De Vlieger, Jul 22 2018 *)

a(n) = A007953(A000668(n)). - Amiram Eldar, Oct 16 2024

Definition corrected by Omar E. Pol, Apr 01 2008

a(38)-a(47) from Gord Palameta, Jul 21 2018

A081093 a(n) is the smallest prime such that the number of 1's in its binary expansion is equal to the n-th prime.

Original entry on oeis.org

3, 7, 31, 127, 3583, 8191, 131071, 524287, 14680063, 1073479679, 2147483647, 266287972351, 4260607557631, 17591112302591, 246290604621823, 17996806323437567, 1152917106560335871, 2305843009213693951

Offset: 1

Reinhard Zumkeller, Mar 05 2003

a(n) = Min{p: A000120(p)=A000040(n), p prime}.
If 2^(Prime[n]) - 1 is a prime number, then a(n) = 2^(Prime[n]) - 1, where Prime[n] denotes the n-th prime number. This means that every Mersenne prime arises in this sequence. - Stefan Steinerberger, Jan 22 2006
For all n with prime(n) < 300, a(n) has either prime(n) or prime(n)+1 bits. - David Wasserman, Oct 25 2006

			n=4, p[4]=11, 3583=[11011111111] has 11 digits=1 and is prime;
2047=23.89=[11111111111] is not here because it is composite.
a(5)=3583=A081092(266)=A000040(502) having eleven 1's: '110111111111' and A000120(p)<11=prime(5) for primes p<3583.
Mersenne-primes are here, Mersenne composites not.

Cf. A000043, A000668, A001348, A061712, A000120, A014499.

Cf. A000040, A000120, A081092.

Do[k=1;While[Count[IntegerDigits[Prime[k], 2], 1] !=Prime[n], k++ ];Print[Prime[k]], {n, 1, 10}]

a(n) = A061712(A000040(n)). - Franklin T. Adams-Watters, Jun 06 2006

More terms from Franklin T. Adams-Watters, Jun 06 2006
Further terms from David Wasserman, Oct 25 2006
Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar

A089158 Second prime factor, if it exists, of Mersenne numbers.

Original entry on oeis.org

89, 178481, 1103, 616318177, 164511353, 9719, 4513, 69431, 3203431780337, 761838257287, 48544121, 2298041, 202029703, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

Offset: 1

Cino Hilliard, Dec 06 2003

nonn

			The 5th Mersenne number 2^11 - 1 = 23*89 and 89 is the second prime divisor.
The 9th Mersenne number 2^23 - 1 = 47*178481 and 178481 is the second prime divisor.
Notice 23, 89 congruent to 1 mod 11 and 47, 178481 congruent to 1 mod 23.

Amiram Eldar, Table of n, a(n) for n = 1..183
Chris Caldwell, Mersenne Primes: History, Theorems and Lists.

Cf. A001348, A016047, A065341, A089159, A089992, A135978.

mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.

Original entry on oeis.org

3, 7, 31, 127, 23, 89, 8191, 131071, 524287, 47, 178481, 233, 1103, 2089, 2147483647, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 2305843009213693951, 193707721, 761838257287

Offset: 1

Cino Hilliard, Dec 06 2003

All factors of Mersenne numbers 2^p - 1, where p is prime, are == 1 (mod p). See the first Caldwell link for a proof of the statement that if q divides M_p = 2^p-1 then q = 2kp + 1 for some integer k. - Comment corrected by Jonathan Sondow, Dec 29 2016

			The 16th Mersenne number 2^53-1 has the three prime factors 6361, 69431, 20394401.
See tail end of second row in the sequence. Each factor is == 1 (mod 53).
Triangle begins:
  3;
  7;
  31;
  127;
  23, 89;
  8191;
  131071;
  524287;
  47, 178481;
  233, 1103, 2089;
  2147483647;
  223, 616318177;
  13367, 164511353;
  431, 9719, 2099863;
  2351, 4513, 13264529;
  6361, 69431, 20394401;

Max Alekseyev, Rows n = 1..197, flattened (rows 1..167 from Jens Kruse Andersen)
R. P. Brent, New factors of Mersenne numbers.
Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists.
Chris K. Caldwell, The Prime Glossary, Mersenne divisor.
Sam Wagstaff, The Cunningham Project.

Cf. A001348, A003260, A016047, A088863.

Cf. A122094 (sorted version of this list).

row[n_]:=First/@FactorInteger[2^Prime[n]-1]; Array[row,19]//Flatten (* Stefano Spezia, May 03 2024 *)

mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

Definition corrected by Max Alekseyev, Jul 25 2023

A100105 a(n) = 2^prime(n)-prime(n).

Original entry on oeis.org

2, 5, 27, 121, 2037, 8179, 131055, 524269, 8388585, 536870883, 2147483617, 137438953435, 2199023255511, 8796093022165, 140737488355281, 9007199254740939, 576460752303423429, 2305843009213693891

Offset: 1

Jorge Coveiro, Dec 26 2004

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..100

Cf. A001348, A034785.

[2^p - p: p in PrimesUpTo(80)]; // Vincenzo Librandi, Jun 03 2014

seq( (2^ithprime(x)-ithprime(x)),x=1..20);

f[n_]:=(2^Prime[n]-Prime[n]); Array[f,4!,1] (* Vladimir Joseph Stephan Orlovsky, Oct 17 2009 *)

cp's OEIS Frontend

A235709 Least prime p < prime(n) with 2^p - 1 a quadratic residue modulo prime(n), or 0 if such a number does not exist.

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A241573 2^p + 3 where p is prime.

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Sage

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

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A188133 Primes p such that 10p+1 divides 2^p-1.

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A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

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Julia

A066538 Sum of the digits of the n-th Mersenne prime (A000668).

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A081093 a(n) is the smallest prime such that the number of 1's in its binary expansion is equal to the n-th prime.

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Mathematica

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A089158 Second prime factor, if it exists, of Mersenne numbers.

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