A242096 a(n) = (n mod 2) * pi( ceiling(n/2)-1 ), where pi is the prime counting function (A000720).

0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 3, 0, 3, 0, 4, 0, 4, 0, 4, 0, 4, 0, 5, 0, 5, 0, 6, 0, 6, 0, 6, 0, 6, 0, 7, 0, 7, 0, 8, 0, 8, 0, 8, 0, 8, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 10, 0, 10, 0, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 12, 0, 12, 0, 12, 0, 12

Offset

0,8

Comments

Start at 0 on the real line and make a continuous clockwise spiral through the integer points: 0 -> 1 -> -1 -> 2 -> -2 -> 3 -> -3, etc. a(n) gives the number of primes on the line strictly between the point after the n-th step of the spiral and 0 (see example).

Links

Derek Orr, Table of n, a(n) for n = 0..10000

Formula

a(n) = (n mod 2) * pi( ceiling(n/2)-1 ).

Example

Table 1:
  n-th step: |0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
  position:  |0, 1,-1, 2,-2, 3,-3, 4,-4, 5, -5,  6, -6,  7, -7,  8, ...
  a(n):      |0, 0, 0, 0, 0, 1, 0, 2, 0, 2,  0,  3,  0,  3,  0,  4, ...
Example 1: a(5) = 1; After the 5th step we are at position +3 on the line and there is exactly one prime between 3 and 0.
                                    •  •  •
                                 •           •
                               •     • • •     •
                              •    •       •    •
                             •    •   • •   •    •
_.___.___.___.___.___.___.___.___.___•___•___.___.___.___.___.___.___.___._
-9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9
                              •   •     •    •
                               •    • •     •
                                 •        •
                                    •  •
Example 2: a(8) = 0; After the 8th step we are at position -4 on the line and there are no (positive) primes between -4 and 0. In general, a(2n)=0.
                                  •  •  •  •
                               •              •
                             •      •  •  •      •
                            •    •          •     •
                           •    •    • • •    •    •
                          •    •   •       •   •    •
                         •    •   •   • •   •   •    •
_.___.___.___.___.___.___.___.___.___•___•___.___.___.___.___.___.___.___._
-9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9
                     •   •    •   •     •   •    •   •
                          •    •    • •    •    •
                      •    •     •       •     •    •
                       •    •      • • •      •    •
                        •     •            •      •
                          •      • • • • •      •
                            •                 •
                                •  •  •   •

Maple

with(numtheory): A242096:=n->(n mod 2)*pi(ceil(n/2)-1): seq(A242096(n), n=0..100);

Mathematica

Table[Mod[n, 2] PrimePi[Ceiling[n/2] - 1], {n, 0, 100}]

Prog

(PARI) vector(100, n, ((n-1)%2)*primepi(ceil((n-1)/2)-1)) \\ Derek Orr, Aug 14 2014

Crossrefs

Cf. A000720.

Sequence in context: A359495 A053399 A322996 * A284965 A274101 A117773

Adjacent sequences: A242093 A242094 A242095 * A242097 A242098 A242099

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 14 2014

Status

approved