A098855 -id:A098855 - cp's OEIS frontend

A110082 Numbers of the form 2^(m-1)*(4^m+2^m-1) where 4^m+2^m-1 is prime.

5, 38, 284, 2168, 133088, 537394688, 140739635806208, 2361183382172302573568, 151115729703628426969088, 20282409604241966234288777068544, 45671926166590726335069952848216804538059849728

Offset: 1

Farideh Firoozbakht, Aug 03 2005

nonn

This sequence is a subsequence of A110079 namely, if n is in the sequence then sigma(n)=2n-2^d(n) where d(n) is number of positive divisors of n(see comments line of the sequence A110079). Sequence A110080 gives numbers n such that 4^n+2^n-1 is prime.

			2^1299*(4^1300+2^1300-1) is in the sequence because 4^1300+2^1300-1 is prime.

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1

Cf. A110079, A098855.

Do[If[PrimeQ[4^m+2^m-1], Print[2^(m-1)*(4^m+2^m-1)]], {m, 52}]

A172994 a(n), starting at n=4, is the smallest positive integral x with an n-th prime in {x^2k+x^k-1} occurring for k < A096594(n).

Original entry on oeis.org

2, 460724, 610357585, 2096681555, 5351622936, 66, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 4

James G. Merickel, Feb 07 2010

Note that the offset here is 4 for the reason that 10^2k+10^k-1 is prime for k=1 through 3 but not for k=4. This sequence is related to the remarkable occurrence of primes in the sequence 109, 10099, 1000999, etc. Second and third terms from Jens Kruse Andersen (prior to submission).

This sequence is essentially complete: a(k)=2 for k>9 with near certainty. That is, assuming the referenced sequences being compared are correct (and they have been checked), this is absolutely known true through a(25); and the contrary at any later point would be comparable to a return to the origin of a random walk on the line that is biased in one direction and already many 'paces' along in that direction. - James G. Merickel, Apr 16 2014

			a(9)=66 corresponds to the fact that 66^48+66^24-1 is already the 9th prime value of type x^2k+x^k-1 for x=66 (i.e., this surpasses A096594(9)=26, that 10^52+10^26-1 is the 9th prime for the case x=10).

Cf. A096594, A098855.

a(9)-a(28) added by James G. Merickel, Mar 23 2014

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

Original entry on oeis.org

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

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A110082 Numbers of the form 2^(m-1)*(4^m+2^m-1) where 4^m+2^m-1 is prime.

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A172994 a(n), starting at n=4, is the smallest positive integral x with an n-th prime in {x^2k+x^k-1} occurring for k < A096594(n).

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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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