cp's OEIS Frontend

A196697 Number of primes of the form of 2^n +- 2^k +- 1 with 0 <= k < n.

%I A196697 #31 Apr 03 2023 10:36:12
%S A196697 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,
%T A196697 12,18,14,15,11,10,10,18,8,12,11,18,12,23,5,12,13,16,13,22,8,9,16,13,
%U A196697 9,13,14,11,11,10,10,20,15,10,10,13,9,22,11,10,10,12
%N A196697 Number of primes of the form of 2^n +- 2^k +- 1 with 0 <= k < n.
%C A196697 Conjecture: all terms of this sequence are greater than 0.
%C A196697 Conjecture tested holds up to n = 10000.
%C A196697 Terms for all n tend to be small integers.
%C A196697 All Mersenne primes and primes of the forms 3*2^n+-1, 5*2^n+-1, 7*2^n+-1, and 15*2^n+-1 form a subgroup of this type of primes.
%C A196697 A large prime that is explicitly found for this type is 2^1048576 - 2^891232 - 1.
%C A196697 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
%H A196697 Lei Zhou, <a href="/A196697/b196697.txt">Table of n, a(n) for n = 1..10000</a>
%H A196697 Chris Caldwell, ed., <a href="https://t5k.org/primes/page.php?id=101355">2^1048576-2^891232-1</a>
%e A196697 For n=1,
%e A196697   2^1 + 2^0 - 1 = 2^1 - 2^0 + 1 = 2: 1 prime, so a(1)=1.
%e A196697 For n=2,
%e A196697   2^2 - 2^0 - 1 = 2;
%e A196697   2^2 - 2^1 + 1 = 3;
%e A196697   2^2 + 2^1 - 1 = 2^2 - 2^1 + 1 = 5;
%e A196697   2^2 + 2^1 + 1 = 7: 4 primes found, so a(2)=4.
%e A196697 ...
%e A196697 For n=11,
%e A196697   2^11 - 2^5 + 1 = 2017;
%e A196697   2^11 - 2^3 - 1 = 2039;
%e A196697   2^11 + 2^2 + 1 = 2053;
%e A196697   2^11 + 2^4 - 1 = 2063;
%e A196697   2^11 + 2^5 + 1 = 2081;
%e A196697   2^11 + 2^6 - 1 = 2111;
%e A196697   2^11 + 2^6 + 1 = 2113: 7 primes found, so a(11)=7.
%t A196697 Table[c1 = 2^i; cs = {};
%t A196697 Do[c2 = 2^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t A196697   cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t A196697   cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t A196697   cp = c1 - c2 - 1;
%t A196697   If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
%t A196697 Length[cs], {i, 1, 100}]
%o A196697 (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
%Y A196697 Cf. A238900 (least k).
%K A196697 nonn
%O A196697 1,2
%A A196697 _Lei Zhou_, Oct 05 2011
%E A196697 Edited by _Jon E. Schoenfield_, Mar 15 2021