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A196697 Number of primes in the form of 2^n +/- 2^k +/- 1 with 0 <= k < n. 7
1, 4, 5, 6, 7, 9, 7, 11, 10, 12, 7, 12, 8, 12, 9, 14, 11, 19, 13, 22, 7, 9, 11, 16, 4, 8, 9, 7, 12, 18, 14, 15, 11, 10, 10, 18, 8, 12, 11, 18, 12, 23, 5, 12, 13, 16, 13, 22, 8, 9, 16, 13, 9, 13, 14, 11, 11, 10, 10, 20, 15, 10, 10, 13, 9, 22, 11, 10, 10, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: all elements of this sequence are greater than 0.

Conjecture tested holds up to n=10000, as of in b-file.

Terms for all n are tend to be small integers.

All Mersenne primes, 3*2^n+/-1, 5*2^n+/-1, 7*2^n+/-1, 15*2^n+/-1 primes are sub group of this type of primes.

A prime that is explicitly found for this type is 2^1048576-2^891232-1.

I conjecture the contrary: infinitely many elements of this sequence are equal to 0. Probably the first n with a(n) = 0 is less than a million. - Charles R Greathouse IV, Nov 21 2011

LINKS

Lei Zhou, Table of n, a(n) for n = 1..10000

Chris Caldwell, ed., 2^1048576-2^891232-1

EXAMPLE

n=1, 2=2^1+2^0-1=2^1-2^0+1 is prime, so a(1)=1;

n=2, 2=2^2-2^0-1; 3=2^2-2^1+1; 5=2^2+2^1-1=2^2-2^1+1; 7=2^2+2^1+1, four primes found, so a(2)=4;

...

n=11,2017=2^11-2^5+1;2039=2^11-2^3-1;2053=2^11+2^2+1;2063=2^11+2^4-1;2081=2^11+2^5+1;2111=2^11+2^6-1;2113=2^11+2^6+1, 7 primes found, so a(11)=7

MATHEMATICA

Table[c1 = 2^i; cs = {};

Do[c2 = 2^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];

  cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];

  cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];

  cp = c1 - c2 - 1;

  If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];

Length[cs], {i, 1, 100}]

PROG

(PARI) a(n)=my(v=List(), t); for(k=0, n-1, if(isprime(t=2^n-2^k-1), listput(v, t)); if(isprime(t=2^n-2^k+1), listput(v, t)); if(isprime(t=2^n+2^k-1), listput(v, t); if(isprime(t=2^n+2^k+1), listput(v, t)))); #Set(v) \\ Charles R Greathouse IV, Oct 06 2011

CROSSREFS

Cf. A238900 (least k).

Sequence in context: A075341 A143789 A068521 * A068318 A242337 A201739

Adjacent sequences:  A196694 A196695 A196696 * A196698 A196699 A196700

KEYWORD

nonn

AUTHOR

Lei Zhou, Oct 05 2011

STATUS

approved

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Last modified July 23 00:43 EDT 2019. Contains 325228 sequences. (Running on oeis4.)