A211349 -id:A211349 - cp's OEIS frontend

A015950 Numbers k such that k | 4^k + 1.

1, 5, 25, 125, 205, 625, 1025, 2525, 3125, 5125, 8405, 12625, 15625, 25625, 42025, 63125, 78125, 103525, 128125, 168305, 202525, 210125, 255025, 315625, 344605, 390625, 517625, 640625, 841525, 875125, 1012625, 1050625, 1275125

Offset: 1

Robert G. Wilson v

nonn

From Robert Israel, Sep 14 2017: (Start)
All terms except 1 are congruent to 5 mod 20.
If k is a term and prime p | k, then k*p is a term.
All prime factors of terms == 1 (mod 4).
If p is a prime == 1 (mod 4) and the order of 4 (mod p) is 2*m where m is in the sequence, then m*p is in the sequence. (End)

			4^5 + 1 = 1025 and 1025 is divisible by 5, so 5 is a term.

Max Alekseyev, Table of n, a(n) for n = 1..3514 (first 325 terms from Robert Israel)

Cf. A015945, A211349.

Column k=4 of A333429.

[n: n in [1..10^6] | Modexp(4, n, n)+1 eq n]; // Jinyuan Wang, Dec 29 2018

select(n -> 4 &^ n + 1 mod n = 0, [1, seq(i,i=5..10^7,20)]); # Robert Israel, Sep 14 2017

Select[Prepend[20 Range[0, 10^5] + 5, 1], Mod[4^# + 1, #] == 0 &] (* Michael De Vlieger, Dec 31 2018 *)

is_A015950(n) = Mod(4,n)^n == -1; \\ Michel Marcus, Sep 15 2017

A015950_list = [n for n in range(1,10**6) if pow(4,n,n) == n-1] # Chai Wah Wu, Mar 25 2021

A069051 Primes p such that p-1 divides 2^p-2.

Original entry on oeis.org

2, 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367, 42463, 71443, 77659, 95923, 99079, 113779, 117307, 143263, 174763, 175447, 184843, 265483, 304039, 308827

Offset: 1

Benoit Cloitre, Apr 03 2002

nonn

These are the prime values of n such that 2^n == 2 (mod n*(n-1)). - V. Raman, Sep 17 2012
These are the prime values p such that n^(2^(p-1)) is congruent to n or -n (mod p) for all n in Z/pZ, the commutative ring associated with each term. This results follows from Fermat's little theorem. - Philip A. Hoskins, Feb 08 2013
A prime p is in this sequence iff p-1 belongs to A014741. For p>2, this is equivalent to (p-1)/2 belonging to A014945. - Max Alekseyev, Aug 31 2016
From Thomas Ordowski, Nov 20 2018: (Start)
Conjecture: if n-1 divides 2^n-2, then (2^n-2)/(n-1) is squarefree.
Numbers n such that b^n == b (mod (n-1)*n) for every integer b are 2, 3, 7, and 43; i.e., only prime numbers of the form A014117(k) + 1. (End)
These are primes p such that p^2 divides b^(2^p-2) - 1 for every b coprime to p. - Thomas Ordowski, Jul 01 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..3314 (First 553 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A216822, A217468, A211203, A211349, A330382.

a(n)-1 form subsequence of A014741; (a(n)-1)/2 for n>1 forms a subsequence of A014945.

Filtered([1..350000],p->IsPrime(p) and (2^p-2) mod (p-1)=0); # Muniru A Asiru, Dec 03 2018

[p : p in PrimesUpTo(310000) | IsZero((2^p-2) mod (p-1))]; // Vincenzo Librandi, Dec 03 2018

Select[Prime[Range[10000]], Mod[2^# - 2, # - 1] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{2,3},Select[Prime[Range[30000]],PowerMod[2,#,#-1]==2&]] (* Harvey P. Dale, Apr 17 2022 *)

isA069051(p)=Mod(2,p-1)^p==2 && isprime(p); \\ Charles R Greathouse IV, Sep 19 2012

from sympy import prime
for n in range(1,350000):
    if (2**prime(n)-2) % (prime(n)-1)==0:
        print(prime(n)) # Stefano Spezia, Dec 07 2018

a(1) added by Charles R Greathouse IV, Sep 19 2012

A211203 Prime numbers p such that p-1 divides (2^(p-1)+1)*(2^p-2).

Original entry on oeis.org

2, 3, 7, 11, 19, 31, 43, 79, 127, 151, 163, 211, 251, 271, 311, 331, 379, 487, 547, 631, 751, 811, 883, 991, 1051, 1171, 1231, 1459, 1471, 1831, 1951, 1999, 2251, 2311, 2531, 2647, 2731, 2791, 2971, 3079, 3331, 3511, 3631, 3691, 3823, 3943, 4051, 4447, 4651

Offset: 1

Philip A. Hoskins, Feb 06 2013

nonn

This is also the set of primes such that n^(4^(p-1)) is congruent to n or -n modulo p.

Prime p>2 is in this sequence iff (p-1)/2 belongs to A014957. - Max Alekseyev, Dec 26 2017

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..204 from Philip A. Hoskins)

Cf. A069051 (primes p such that p - 1 divides 2^p - 2)

Cf. A211349 (primes p such that p - 1 divides 2^p + 2)

A211203:=proc(q)
local n;
for n from 1 to q do
  if type((2^(2*ithprime(n)-1)-2)/(ithprime(n)-1),integer) then print(ithprime(n));
fi; od; end:
A211203(10000000); # Paolo P. Lava, Feb 18 2013

Select[Prime[Range[1000]], Mod[1/2*(2^# + 2)*(2^# - 2), # - 1] == 0 &]

is(p) = lift((Mod(2,p-1)^(p-1)+1)*(Mod(2,p-1)^p-2))==0 \\ David A. Corneth, Mar 25 2021

from sympy import primerange
A211203_list = [p for p in primerange(1,10**6) if p == 2 or p == 3 or pow(2,2*p-1,p-1) == 2] # Chai Wah Wu, Mar 25 2021

cp's OEIS Frontend

A015950 Numbers k such that k | 4^k + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

A069051 Primes p such that p-1 divides 2^p-2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Python

Extensions

A211203 Prime numbers p such that p-1 divides (2^(p-1)+1)*(2^p-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python