A057182 -id:A057182 - cp's OEIS frontend

A231865 Numbers n such that (43^n + 1)/44 is prime.

5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573

Offset: 1

Robert Price, Nov 14 2013

All terms are primes.

a(11) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604.

Do[ p=Prime[n]; If[ PrimeQ[ (43^p + 1)/44 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((43^n+1)/44) \\ Charles R Greathouse IV, Feb 20 2017

A235683 Numbers n such that (46^n + 1)/47 is prime.

Original entry on oeis.org

7, 23, 59, 71, 107, 223, 331, 2207, 6841, 94841

Offset: 1

Robert Price, Jan 13 2014

All terms up to a(10) are primes.

a(11) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865.

Do[ p=Prime[n]; If[ PrimeQ[ (46^p + 1)/47 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((46^n+1)/47) \\ Charles R Greathouse IV, May 22 2017

A237052 Numbers n such that (49^n + 1)/50 is prime.

Original entry on oeis.org

7, 19, 37, 83, 1481, 12527, 20149

Offset: 1

Robert Price, Feb 02 2014

All terms are primes.

a(8) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime.

Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683, A236167, A236530.

Do[ p=Prime[n]; If[ PrimeQ[ (49^p + 1)/50 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((49^n+1)/50) \\ Charles R Greathouse IV, Jun 13 2017

Typo in description corrected by Ray Chandler, Feb 20 2017

A283657 Numbers m such that 2^m + 1 has at most 2 distinct prime factors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479

Offset: 1

Vladimir Shevelev, Mar 13 2017

nonn

Using comment in A283364, note that if a(n) is odd > 9, then it is prime.
503 <= a(41) <= 596. - Robert Israel, Mar 13 2017
Could (4^p + 1)/5^t be prime, where p is prime, 5^t is the highest power of 5 dividing 4^p + 1, other than for p=2, 3 and 5? - Vladimir Shevelev, Mar 14 2017
In his message to seqfans from Mar 15 2017, Jack Brennen beautifully proved that there are no more primes of such form. From his proof one can see also that there are no terms of the form 2*p > 10 in the sequence. - Vladimir Shevelev, Mar 15 2017
Where A046799(n)=2. - Robert G. Wilson v, Mar 15 2017
From Giuseppe Coppoletta, May 16 2017: (Start)
The only terms that are not in A066263 are those m giving 2^m + 1 = prime (i.e. m = 0 and any number m such that 2^m + 1 is a Fermat prime) and the values of m giving 2^m + 1 = power of a prime, giving m = 3 as the only possible case (by Mihăilescu-Catalan's result, see links).
For the relation with Fermat numbers and for other possible terms to check, see comments in A073936 and A066263.
All terms after a(59) refer to probabilistic primality tests for 2^a(n) + 1  (see Caldwell's link for the list of the largest certified Wagstaff primes).
After a(65), the values 267017, 269987, 374321, 986191, 4031399 and 4101572 are also terms, but there still remains the remote possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but possibly much further along in the numbering (see comments in A000978).
(End).

			0 is a term as 2^0 + 1 = 2 is a prime.
10 is a term as 2^10 + 1 = 5^2 * 41.
14 is not a term as 2^14 + 1 = 5 * 29 * 113.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..65
Jack Brennen, Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017).
C. Caldwell's The Top Twenty Wagstaff primes.
Mersennewiki, Factorizations Of Cunningham Numbers C+(2,n) (tables).
Samuel S. Wagstaff, The Cunningham Project.
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Eric Weisstein's World of Mathematics, Zsigmondy Theorem.

Cf. A002587, A046799, A283364, A073936, A000978, A127317, A092559, A019434, A066263.

Contains 4*A057182.

# this uses A002587[i] for i<=500, e.g., from the b-file for that sequence
count:= 0:
for i from 0 to 500 do
  m:= 0;
  r:= (2^i+1);
  if i::odd then
    m:= 1;
    r:= r/3^padic:-ordp(r,3);
  elif i > 2 then
    q:= max(numtheory:-factorset(i));
    if q > 2 then
      m:= 1;
      r:= r/B[i/q]^padic:-ordp(r,A002587[i/q]);
    fi
  fi;
  if r mod B[i] = 0 then m:= m+1;
      j:= padic:-ordp(r, A002587[i]);
      r:= r/B[i]^j;
  fi;
  mmax:= m;
  if isprime(r) then m:= m+1; mmax:= m
  elif r > 1 then mmax:= m+2
  fi;
  if mmax <= 2 or (m <= 1 and m + nops(numtheory:-factorset(r)) <= 2) then
       count:= count+1;
     A[count]:= i;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Mar 13 2017

Select[Range[0, 313], PrimeNu[2^# + 1]<3 &] (* Indranil Ghosh, Mar 13 2017 *)

for(n=0, 313, if(omega(2^n + 1)<3, print1(n,", "))) \\ Indranil Ghosh, Mar 13 2017

a(16)-a(38) from Peter J. C. Moses, Mar 13 2017
a(39)-a(40) from Robert Israel, Mar 13 2017
a(41)-a(65) from Giuseppe Coppoletta, May 08 2017

A309533 Numbers k such that (144^k + 1)/145 is prime.

Original entry on oeis.org

23, 41, 317, 3371, 45259, 119671

Offset: 1

Paul Bourdelais, Aug 06 2019

The corresponding primes are terms of A059055. - Bernard Schott, Aug 09 2019

Cf. A000978, A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A126856, A185240.

Do[p=Prime[n]; If[PrimeQ[(144^p + 1)/145], Print[p]], {n, 1, 1000000}]

PARI
```
is(n)=ispseudoprime((144^n+1)/145)
```

A236167 Numbers k such that (47^k + 1)/48 is prime.

Original entry on oeis.org

5, 19, 23, 79, 1783, 7681

Offset: 1

Robert Price, Jan 19 2014

a(7) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field.
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers k such that (2^k + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683.

Do[ p=Prime[n]; If[ PrimeQ[ (47^p + 1)/48 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((47^n+1)/48) \\ Charles R Greathouse IV, Jun 06 2017

from sympy import isprime
def afind(startat=0, limit=10**9):
  pow47 = 47**startat
  for k in range(startat, limit+1):
    q, r = divmod(pow47+1, 48)
    if r == 0 and isprime(q): print(k, end=", ")
    pow47 *= 47
afind(limit=300) # Michael S. Branicky, May 19 2021

A185230 Numbers n such that (33^n + 1)/34 is prime.

Original entry on oeis.org

5, 67, 157, 12211, 313553

Offset: 1

Robert Price, Aug 29 2013

All terms are prime.

a(5) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382. Cf. A084741, A084742, A065507, A126659, A126856.

Do[ p=Prime[n]; If[ PrimeQ[ (33^p + 1)/34 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((33^n+1)/34) \\ Charles R Greathouse IV, Jun 13 2017

a(5) from Paul Bourdelais, Feb 26 2021

A195439 Numbers n such that 16^n + 1 is a semiprime.

Original entry on oeis.org

3, 5, 7, 8, 10, 16, 23, 26, 32, 37, 64, 89, 149, 173, 251, 307, 317, 956, 30197, 46058

Offset: 1

Arkadiusz Wesolowski, Oct 19 2011

nonn

Cf. A002590, A057182, A127317.

Select[Range[37], PrimeOmega[16^# + 1] == 2 &] (* Arkadiusz Wesolowski, Dec 15 2011 *)

A236530 Numbers n such that (48^n + 1)/49 is prime.

Original entry on oeis.org

5, 17, 131, 84589

Offset: 1

Robert Price, Jan 27 2014

All terms are primes.

a(5) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683, A236167.

Do[ p=Prime[n]; If[ PrimeQ[ (48^p + 1)/49 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((48^n+1)/49) \\ Charles R Greathouse IV, Jun 13 2017

Incorrect first term deleted by Robert Price, Feb 21 2014

A347138 Numbers k such that (100^k + 1)/101 is prime.

Original entry on oeis.org

3, 293, 461, 11867, 90089

Offset: 1

Paul Bourdelais, Aug 19 2021

These are the repunit primes in base -100. It is unusual to represent numbers in a negative base, but it follows the same formulation as any base: numbers are represented as a sum of powers in that base, i.e., a0*1 + a1*b^1 + a2*b^2 + a3*b^3 ... Since the base is negative, the terms will be alternating positive/negative. For repunits the coefficients are all ones so the sum reduces to 1 + b + b^2 + b^3 + ... + b^(k-1) = (b^k-1)/(b-1). Since b is negative and k is an odd prime, the sum equals (|b|^k+1)/(|b|+1). For k=3, the sum is 9901, which is prime. As with all repunits, we only need to PRP test the prime exponents. The factors of repunits base -100 will be of the form p=2*k*m+1 where m must be even, which is common for (negative) bases that are squares.

			3 is a term since (100^3 + 1)/101 = 9901 is a prime.

Paul Bourdelais, A Generalized Repunit Conjecture

Cf. A309533, A309532, A237052, A229145, A057191, A057182, A057175.

Do[ If[ PrimeQ[ (100^n + 1)/101], Print[n]], {n, 0, 18000}]

PARI
```
is(n)=isprime((100^n+1)/101)
```

cp's OEIS Frontend

A231865 Numbers n such that (43^n + 1)/44 is prime.

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A235683 Numbers n such that (46^n + 1)/47 is prime.

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A237052 Numbers n such that (49^n + 1)/50 is prime.

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A283657 Numbers m such that 2^m + 1 has at most 2 distinct prime factors.

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A309533 Numbers k such that (144^k + 1)/145 is prime.

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A236167 Numbers k such that (47^k + 1)/48 is prime.

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A185230 Numbers n such that (33^n + 1)/34 is prime.

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A195439 Numbers n such that 16^n + 1 is a semiprime.

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