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A162203
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The mountain path of the primes (see comment lines for definition).
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12
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2, 2, 2, 3, 1, -1, 1, 3, 1, -1, 1, 3, 1, -3, 1, 4, 1, -2, 1, 5, 1, -1, 1, 3, 1, -3, 1, 6, 1, -2, 1, 4, 1, -3, 1, 3, 1, -2, 1, 5, 1, -3, 1, 7, 1, -4, 1, 3, 1, -1, 1, 3, 1, -1, 1, 9, 1, -7, 1, 5, 1, -2, 1, 6, 1, -4, 1, 4, 1, -4, 1, 5, 1, -3, 1, 6, 1, -2, 1, 6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| On the infinite square grid we draw a infinite straight line from the point (1,0) in direction (2,1).
We start at stage 1 from the point (0,0) drawing a edge ((0,0),(2,0)) in a horizontal direction.
At stage 2 we draw a edge ((2,0),(2,2)) in a vertical direction. We can see that the straight line intercepts to the number 3 (The first odd prime).
At stage 3 we draw a edge ((2,2),(4,2)) in a horizontal direction. We can see that the straight line intercepts to the number 5 (The second odd prime).
And so on (see illustrations).
The absolute value of a(n) is equal to the length of the n-th edge of a path, or infinite square polyedge, such that the mentioned straight line intercepts, on the path, to the number 1 and the odd primes. In other words, the straight line intercepts the odd non-composite numbers (A006005).
The position of the x-th odd non-composite number A006005(x) is represented by the point P(x,x-1) .
So the position of the first prime number is represented by the point P(2,0) and position of the x-th prime A000040(x), for x>1, is represented by the point P(x,x-1), for example: 31, the 11-st prime, is represented by the point P(11,10).
See also A162200, A162201 and A162202 for more information.
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LINKS
| O. E. Pol, Graph of the mountain path function for prime numbers
O. E. Pol, Illustration: The mountain path of the primes
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FORMULA
| Contribution from Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), May 10 2011: (Start)
a(2n+1) = 1 for n >= 2.
a(2n) = (-1)^n*(A162341(n+2) - 1) = (-1)^n*(A052288(n) - 1) + 1 for n >= 2. (End)
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EXAMPLE
| Array begins:
=====
X..Y
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2, 2;
2, 3;
1,-1;
1, 3;
1,-1;
1, 3;
1,-3;
1, 4;
1,-2;
1, 5;
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CROSSREFS
| Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162340, A162341, A162342, A162343, A162344.
Sequence in context: A004489 A112599 A106795 * A071455 A198862 A139465
Adjacent sequences: A162200 A162201 A162202 * A162204 A162205 A162206
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KEYWORD
| easy,sign
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AUTHOR
| Omar E. Pol (info(AT)polprimos.com), Jun 27 2009
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EXTENSIONS
| Edited by Omar E. Pol (info(AT)polprimos.com), Jul 02 2009
More terms from Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), May 10 2011
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