A196697 Number of primes of the form of 2^n +- 2^k +- 1 with 0 <= k < n.
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
Offset: 1
Keywords
Examples
For n=1, 2^1 + 2^0 - 1 = 2^1 - 2^0 + 1 = 2: 1 prime, so a(1)=1. For n=2, 2^2 - 2^0 - 1 = 2; 2^2 - 2^1 + 1 = 3; 2^2 + 2^1 - 1 = 2^2 - 2^1 + 1 = 5; 2^2 + 2^1 + 1 = 7: 4 primes found, so a(2)=4. ... For n=11, 2^11 - 2^5 + 1 = 2017; 2^11 - 2^3 - 1 = 2039; 2^11 + 2^2 + 1 = 2053; 2^11 + 2^4 - 1 = 2063; 2^11 + 2^5 + 1 = 2081; 2^11 + 2^6 - 1 = 2111; 2^11 + 2^6 + 1 = 2113: 7 primes found, so a(11)=7.
Links
- Lei Zhou, Table of n, a(n) for n = 1..10000
- Chris Caldwell, ed., 2^1048576-2^891232-1
Crossrefs
Cf. A238900 (least k).
Programs
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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}] -
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
Extensions
Edited by Jon E. Schoenfield, Mar 15 2021
Comments