A249759 -id:A249759 - cp's OEIS frontend

A268212 Numbers n of the form 2^k + 1 such that n^2 - n - 1 is a prime q (for k >= 0).

3, 5, 9, 17, 65, 1025, 65537, 16777217, 67108865, 34359738369, 4503599627370497, 36028797018963969, 39614081257132168796771975169, 22300745198530623141535718272648361505980417

Offset: 1

Jaroslav Krizek, Jan 28 2016

nonn

Conjecture: subsequence of prime terms (3, 5, 17, 65537, ...) is not the same as A249759.
Corresponding values of numbers k are in A098855 (numbers n such that 4^n + 2^n - 1 is prime).
Corresponding values of primes q: 5, 19, 71, 271, 4159, 1049599, 4295032831, ...
4 out of 5 known Fermat primes (3, 5, 17, 65537) are terms; corresponding values of primes q: 5, 19, 271, 4295032831.

			17  = 2^4 + 1 is a term because 17^2 - 17 - 1 = 271 (prime).

Intersection of A002328 and A000051.

Cf. A019434, A091567, A098855, A249759.

[2^n + 1: n in [0..300] | IsPrime((2^n + 1)^2 - 2^n - 2)]

2^# + 1 &@ Select[Range[0, 300], PrimeQ[#^2 - # - 1 &@ (2^# + 1)] &] (* Michael De Vlieger, Jan 29 2016 *)

lista(nn) = {for (k=0, nn, n = 2^k+1; if (isprime(n^2-n-1), print1(n, ", ")););} \\ Michel Marcus, Mar 06 2016

A281622 Numbers k such that sigma(k-1) is a Mersenne prime (A000668).

Original entry on oeis.org

3, 5, 17, 26, 65, 4097, 65537, 262145, 1073741825

Offset: 1

Jaroslav Krizek, Jan 25 2017

Conjecture 1: the next terms are: 1152921504606846977, 309485009821345068724781057, 81129638414606681695789005144065, 85070591730234615865843651857942052865.
Conjecture 2: Union of 26 and A256438.
Conjecture 3: Mersenne prime 31 is the only prime p such that p = sigma(x-1) = sigma(y-1) for distinct numbers x and y; 31 = sigma(17-1) = sigma(26-1).

			65 is a term because sigma(64) = 127 (Mersenne prime).

Union of 26 and odd terms of A270413.
Prime terms are in A249759.
Subsequence of A270413.
Cf. A000203, A000668, A256438.

[n: n in[2..1000000], k in [1..20] | SumOfDivisors(n-1) eq 2^k-1 and IsPrime(2^k-1)];

isok(n) = my(s = sigma(n-1)); isprime(s) && ispower(s+1,,&p) && (p==2); \\ Michel Marcus, Jan 27 2017

Conjecture: a(n) = 2^A090748(n) + 1. - Daniel Suteu, Feb 08 2017

cp's OEIS Frontend

A268212 Numbers n of the form 2^k + 1 such that n^2 - n - 1 is a prime q (for k >= 0).

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