A173752 a(n) = m-k where (prime(k), prime(m)) is the n-th prime pair (x^2-x+11, x^2+x+11), integer x >= 0.
0, 1, 1, 2, 2, 2, 3, 3, 4, 3, 5, 7, 5, 5, 9, 8, 10, 17, 15, 15, 16, 15, 17, 18, 20, 23, 27, 25, 30, 27, 26, 30, 32, 39, 43, 49, 48, 55, 54, 48, 64, 66, 62, 61, 62, 68, 63, 65, 77, 65, 73, 79, 85, 73, 86, 93, 98, 84, 100, 107, 113, 110, 105, 107, 121, 119, 120, 119, 121, 125, 114
Offset: 1
Keywords
Examples
The first prime pair (x^2-x+11, x^2+x+11) is obtained for x = 0: 0^2-0+11 = 11 and 0^2+0+11 = 11; 11 is the fifth prime, hence a(1) = 5-5 = 0. The second prime pair is obtained for x = 1: 1^2-1+11 = 11 and 1^2+1+11 = 13; 11 is the fifth prime and 13 is the sixth prime, hence a(2) = 6-5 = 1. The third prime pair is obtained for x = 2: 2^2-2+11 = 13 and 2^2+2+11 = 17; 13 is the sixth prime and 17 is the seventh prime, hence a(3) = 7-6 = 1. The eleventh prime pair is obtained for x = 13: 13^2-13+11 = 167 and 13^2+13+11 = 193; 167 is prime(39) and 193 is prime(44), hence a(11) = 44-39 = 5.
Links
- Harvey P. Dale, Table of n, a(n) for n = 1..1000
Crossrefs
Programs
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Magma
PrimePi:=func< n | #PrimesUpTo(n) >; [ PrimePi(p)-PrimePi(q): x in [0..850] | IsPrime(p) and IsPrime(q) where p is x^2+x+11 where q is x^2-x+11 ]; // Klaus Brockhaus, Feb 27 2010
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Maple
for x from 0 to 1000 do mp := x^2+x+11 ; kp := x^2-x+11 ; if isprime(mp) and isprime(kp) then m := numtheory[pi](mp) ; k := numtheory[pi](kp) ; printf("%d,",m-k) ; end if; end do : # R. J. Mathar, Mar 01 2010 -
Mathematica
pp[n_]:=Module[{c=n^2+11},If[AllTrue[c+{n,-n},PrimeQ],PrimePi[c+n]- PrimePi[ c-n],0]]; Join[{0},Array[pp,1000]/.(0->Nothing)] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 15 2017 *)
Extensions
Edited and extended by Klaus Brockhaus, Feb 27 2010
a(15) corrected and sequence extended by R. J. Mathar, Mar 01 2010
Comments