A057726 -id:A057726 - cp's OEIS frontend

A057727 3^(2^n) + 2.

5, 11, 83, 6563, 43046723, 1853020188851843, 3433683820292512484657849089283, 11790184577738583171520872861412518665678211592275841109096963

Offset: 0

G. L. Honaker, Jr., Oct 28 2000

Cf. A057726.

3^2^Range[0,10]+2 (* Harvey P. Dale, Jan 01 2021 *)

A286680 Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime.

Original entry on oeis.org

0, 5, 4, 2, 0, 3, 1, 0, 3, 3, 0, 1, 0, 0, 2, 4, 0, 0, 2, 0, 2, 1, 0, 2, 0, 0, 1, 0, 0, 2, 3, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2

Offset: 0

Juri-Stepan Gerasimov, May 12 2017

nonn

Nonprimes: 1, 4294967297, 43046723, 259, 9, 1679621, 55, 15, 43046729, 100000009, 21, 155, 25, 27, 50639, 18446744073709551631, 33, 35, ...
Conjecture: a(n) <= 6 for all n.
This conjecture would contradict the generalized Bunyakovsky conjecture.  That is, the polynomials (1+n)^k+n for k=0..6 satisfy the conditions for that conjecture, and so there should be some n for which all seven are prime. - Robert Israel, May 17 2017
Smallest k such that (1 + k)^(2^n) + k is not prime: 0, 6, 3, 5, 2, 1, 54131988 (conjecturally finite). Last term found by Robert G. Wilson v, May 14 2017
From Robert G. Wilson v, May 18 2017: (Start)
m=
0: 0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, etc.;
1: 6, 11, 21, 26, 33, 35, 36, 39, 41, 48, 50, 51, 56, 68, 74, 78, 81, 83, etc.;
2: 3, 14, 18, 20, 23, 29, 44, 54, 63, 65, 69, 75, 95, 99, 113, 114, 125, etc.;
3: 5, 8, 9, 30, 53, 119, 230, 308, 329, 350, 624, 638, 779, 785, 813, 1110, etc.;
4: 2, 15, 2100, 4223, 4773, 7868, 8744, 9339, 9540, 13178, 14589, 15884, etc.;
5: 1, 1432578, 1627035, 1737054, 1888094, 1959638, 2176139, 3172304, 3425069, etc.;
6: 54131988, 177386619, 229940778, 846372674, 2124404844, 2367307088, 2539775055, etc.;
(End)

			a(0) = 0 because (1 + 0)^(2^0) + 0 = 1 is not prime.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Wikipedia, Bunyakovsky conjecture.

Cf. A019434, A047845, A057726, A160027.

f:= proc(n) local k;
  for k from 0 while isprime((1+n)^(2^k)+n) do od:
  k;
end proc:
map(f, [$0..100]); # Robert Israel, May 17 2017

f[n_] := Block[{k = 0}, While[ PrimeQ[(1 + n)^(2^k) + n], k++]; k]; Array[f, 105, 0] (* Robert G. Wilson v, May 14 2017 *)

a(n) = {my(m = 0); while (isprime((1 + n)^(2^m) + n), m++); m;} \\ Michel Marcus, May 19 2017

A152589 Primes of the form 15^(2^k) + 2.

Original entry on oeis.org

17, 227, 50627, 6568408355712890627

Offset: 1

Cino Hilliard, Dec 08 2008

nonn

Cf. A057726, A152588.

g(a,n) = if(a%2, b=2, b=1); for(x=0, n, y=a^(2^x)+b; if(ispseudoprime(y), print1(y, ", ")))
g(15, 4)

A286982 Smallest nonnegative k such that (1 + k)^(2^n) + k is not prime and all (1 + k)^(2^j) + k, for 0 <= j < n, are primes.

Original entry on oeis.org

6, 3, 5, 2, 1, 54131988

Offset: 1

Juri-Stepan Gerasimov, May 12 2017

			a(1) = 6 because (1 + 6)^(2^1) + 6 = 55 is semiprime and (1 + 6)^(2^0) + 6 = 13 is prime;
a(2) = 3 because (1 + 3)^(2^2) + 3 = 259 is semiprime and both (1 + 3)^(2^0) + 3 = 7 and (1 + 3)^(2^1) + 3 = 19 are primes;
a(3) = 5 because (1 + 5)^(2^3) + 5 = 167921 is semiprime and (1 + 5)^(2^0) + 5 = 11, (1 + 5)^(2^1) + 5 = 41 and (1 + 5)^(2^2) + 5 = 1301 are all primes;
a(4) = 2 because (1 + 2)^(2^4) + 2 = 43046723 is semiprime and (1 + 2)^(2^0) + 2 = 5, (1 + 2)^(2^1) + 2 = 11, (1 + 2)^(2^2) + 2 = 83 and (1 + 2)^(2^3) + 2 = 6563 are all primes;
a(5) = 1 because (1 + 1)^(2^5) + 1 = 4294967297 is semiprime and (1 + 1)^(2^0) + 1 = 3, (1 + 1)^(2^1) + 1 = 5, (1 + 1)^(2^2) + 1 = 17, (1 + 1)^(2^3) + 1 = 257 and (1 + 1)^(2^4) + 1 = 65537 are fix known Fermat primes (A019434);
a(6) = 54131988 because (1 + 54131988)^(2^6) + 54131988 is composite and (1 + 54131988)^(2^0) + 54131988 = 108263977, (1 + 54131988)^(2^1) + 54131988 = 2930272287228109, (1 + 54131988)^(2^2) + 54131988 =  8586495360054127683625679378629, (1 + 54131988)^(2^3) + 54131988 = 73727902568231063808600888120898279950965368674840612135914869, (1 + 54131988)^(2^4) + 54131988 and (1 + 54131988)^(2^5) + 54131988 are all primes.

Cf. A019434, A047845, A057726, A160027, A286680.

a[n_] := Block[{k = 1}, While[PrimeQ[(1 + k)^(2^n) + k] || ! AllTrue[(1 + k)^(2^Range[0, n-1]) + k, PrimeQ], k++]; k]; Array[a, 5] (* Giovanni Resta, May 30 2017 *)

a(6) from Robert G. Wilson v, May 14 2017

A174262 Primes of the form 21^(2^n) + 2.

Original entry on oeis.org

23, 443, 194483

Offset: 1

Giovanni Teofilatto, Mar 14 2010

a(n) is congruent 3 mod 4.

The next term (if it exists) corresponds to some n>23, so it has more than 11 million digits. [R. J. Mathar, Aug 23 2010]

Cf. A057726, A152589.

Entries checked. - R. J. Mathar, Aug 23 2010

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A057727 3^(2^n) + 2.

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A286680 Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime.

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A152589 Primes of the form 15^(2^k) + 2.

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A286982 Smallest nonnegative k such that (1 + k)^(2^n) + k is not prime and all (1 + k)^(2^j) + k, for 0 <= j < n, are primes.

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A174262 Primes of the form 21^(2^n) + 2.

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