A162203 -id:A162203 - cp's OEIS frontend

A162200 Number on the positive y axis of the n-th horizontal component in the graph of the "mountain path" function for prime numbers.

0, 0, 2, 2, 5, 4, 7, 6, 9, 6, 10, 8, 13, 12, 15, 12, 18, 16, 20, 17, 20, 18, 23, 20, 27, 23, 26, 25, 28, 27, 36, 29, 34, 32, 38, 34, 38, 34, 39, 36, 42, 40, 46, 42, 45, 44, 51, 41, 49, 48, 51, 48, 52, 48, 56, 52, 58, 56, 60, 57, 60, 56, 68, 61, 64, 63, 72, 64, 72, 68, 71, 68, 75

Offset: 1

Omar E. Pol, Jun 28 2009

Note that the n-th horizontal component is an edge with length equal to 1 (see the link: Graph of the mountain path function).

See A162201 for the first differences.

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A000040, A006005, A008578, A162201, A162202, A162203, A162340, A162341, A162342, A162343, A162344.

A031131 := proc(n) ithprime(n+2)-ithprime(n) ; end: A052288 := proc(n) A031131(n)/2 ; end: A162201 := proc(n) if n <= 2 then 2*(n-1); elif n mod 2 = 0 then A052288(n-1) ; else 2-A052288(n-1) ; fi; end: A162200 := proc(n) add(A162201(j),j=1..n-1) ; end: seq(A162200(n),n=1..150) ; # R. J. Mathar, Jul 15 2009

Edited by Omar E. Pol, Jul 02 2009

More terms from R. J. Mathar, Jul 15 2009

A162201 First differences of A162200.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 28 2009

The absolute value of a(n) is also the length of the n-th vertical edge in the graph of the "mountain path" function for prime numbers.

See A162200 for the length of the n-th horizontal component.

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A000040, A006005, A008578, A162200, A162202, A162203, A162340, A162341, A162342, A162343, A162344.

A031131 := proc(n) ithprime(n+2)-ithprime(n) ; end: A052288 := proc(n) A031131(n)/2 ; end: A160173 := proc(n) if n <= 2 then 2*(n-1); elif n mod 2 = 0 then A052288(n-1) ; else 2-A052288(n-1) ; fi; end: seq(A160173(n),n=1..150) ; # R. J. Mathar, Jul 15 2009

From R. J. Mathar, Jul 15 2009: (Start)
a(n) = A052288(n-1) if n >= 2, n even.
a(n) = 2 - A052288(n-1) if n >= 3, n odd. (End)

Edited by Omar E. Pol, Jul 02 2009

More terms from R. J. Mathar, Jul 15 2009

A162202 Number of the n-th vertex in the graph of the "mountain path" function for prime numbers.

Original entry on oeis.org

0, 2, 4, 6, 9, 10, 11, 12, 15, 16, 17, 18, 21, 22, 25, 26, 30, 31, 33, 34, 39, 40, 41, 42, 45, 46, 49, 50, 56, 57, 59, 60, 64, 65, 68, 69, 72, 73, 75, 76, 81, 82, 85, 86, 93, 94, 98, 99, 102, 103, 104, 105, 108, 109, 110, 111, 120, 121, 128, 129, 134, 135

Offset: 0

Omar E. Pol, Jun 28 2009

This sequence is formed by a zero together with the partial sums of the absolute values of A162203.

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A000040, A006005, A008578, A162200, A162201, A162203, A162340, A162341, A162342, A162343, A162344.

Edited by Omar E. Pol, Jul 02 2009

More terms from Nathaniel Johnston, May 10 2011

A162340 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the graph of the "mountain path" function for prime numbers.

Original entry on oeis.org

1, 2, 5, 6, 10, 12, 16, 18, 22, 26, 31, 34, 40, 42, 46, 50, 57, 60, 65, 69, 73, 76, 82, 86, 94, 99, 103, 105, 109, 111, 121, 129, 135, 138, 145, 150, 155, 160, 166, 170, 177, 180, 187, 192, 196, 198, 206, 217, 226, 228, 232, 236, 241, 246, 255, 260, 267, 270, 275, 279

Offset: 0

Omar E. Pol, Jul 01 2009

nonn

a(n) is also the number of grid points that are covered after n-th stage by the same polyedge mentioned in the definition of this sequence.

Also, partial sums of A162341.

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162203, A162341.

Edited by Omar E. Pol, Jul 05 2009

More terms from Nathaniel Johnston, Nov 06 2010

A162341 a(n) = number of grid points P(x,y) that are covered by a polyedge as the graph of the "mountain path" function for prime numbers, where x=n and y=0..oo.

Original entry on oeis.org

1, 1, 3, 1, 4, 2, 4, 2, 4, 4, 5, 3, 6, 2, 4, 4, 7, 3, 5, 4, 4, 3, 6, 4, 8, 5, 4, 2, 4, 2, 10, 8, 6, 3, 7, 5, 5, 5, 6, 4, 7, 3, 7, 5, 4, 2, 8, 11, 9, 2, 4, 4, 5, 5, 9, 5, 7, 3, 5, 4, 4, 5, 13, 8, 4, 2, 10, 9, 9, 5, 4, 4, 8, 6, 7, 4, 6, 6, 7, 5, 10, 5, 7, 5, 5, 4, 6, 6, 7, 2, 4, 7, 11, 5, 7, 5, 6, 8, 8, 9, 13, 7, 9

Offset: 0

Omar E. Pol, Jul 01 2009

nonn

Se also A162340.

Omar E. Pol, Graph of the mountain path function for prime numbers

Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162203, A162340.

a(n) = A052288(n-2) + (-1)^n for n>=3. [From Nathaniel Johnston, Nov 06 2010]

Edited by Omar E. Pol, Jul 05 2009

More terms from Nathaniel Johnston, Nov 06 2010

A162342 Partial sums of A162200.

Original entry on oeis.org

0, 0, 2, 4, 9, 13, 20, 26, 35, 41, 51, 59, 72, 84, 99, 111, 129, 145, 165, 182, 202, 220, 243, 263, 290, 313, 339, 364, 392, 419, 455, 484, 518, 550, 588, 622, 660, 694, 733, 769, 811, 851, 897, 939, 984, 1028, 1079, 1120, 1169, 1217, 1268, 1316, 1368, 1416

Offset: 0

Omar E. Pol, Jul 01 2009

Cf. A162200, A162201, A162202, A162203, A162340, A162341.

More terms from Nathaniel Johnston, May 10 2011

A162343 Array read by rows in which row n lists the numbers that are in the same "y" level in the mountain path of the primes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 9, 10, 13, 14, 17, 18, 25, 26, 15, 16, 19, 24, 27, 20, 23, 28, 33, 34, 21, 22, 29, 32, 35, 30, 31, 36, 37, 38, 41, 42, 49, 50, 39, 40, 43, 48, 51, 44, 47, 52, 45, 46, 53, 54, 59, 60, 55, 58, 61, 68, 69, 56, 57, 62, 67, 70, 75, 76, 63, 66, 71, 74

Offset: 0

Omar E. Pol, Jul 24 2009

See the illustration "The mountain path of the primes", here.

			Array begins:
0, 1, 2,. . . . . . . . . . . . . . . . . .
. . . 3,. . . . . . . . . . . . . . . . . .
. . . 4, 5, 6,. . . . . . . . . . . . . . .
. . . . . . 7,. . . . . . . . . . . . . . .
. . . . . . 8,11,12,. . . . . . . . . . . .
. . . . . . 9,10,13,. . . . . . . . . . . .
. . . . . . . . .14,17,18,25,26,. . . . . .
. . . . . . . . .15,16,19,24,27,. . . . . .
. . . . . . . . . . . .20,23,28,33,34,. . .
. . . . . . . . . . . .21,22,29,32,35,. . .
. . . . . . . . . . . . . . ,30,31,36,. . .
. . . . . . . . . . . . . . . . . .37,. . .

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A162200, A162201, A162202, A162203, A162340, A162341, A162342, A162344, A162345, A162350, A162351, A162352

A162344 Array read by rows in which row n lists the numbers that are in the same "x" level in the mountain path of the primes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 25, 24, 23, 22, 26, 27, 28, 29, 30, 33, 32, 31, 34, 35, 36, 37, 38, 39, 41, 40, 42, 43, 44, 45, 49, 48, 47, 46, 50, 51, 52, 53, 54, 55, 56, 59, 58, 57, 60, 61, 62, 63, 64, 68, 67, 66, 65, 69, 70, 71, 72

Offset: 0

Omar E. Pol, Jul 24 2009

See the illustration "The mountain path of the primes", here.

			Array begins:
0, 1, 2,. . . . . . . . . . . . . . . . . .
. . . 3,. . . . . . . . . . . . . . . . . .
. . . 4, 5, 6,. . . . . . . . . . . . . . .
. . . . . . 7,. . . . . . . . . . . . . . .
. . . . . . 8,11,12,. . . . . . . . . . . .
. . . . . . 9,10,13,. . . . . . . . . . . .
. . . . . . . . .14,17,18,25,26,. . . . . .
. . . . . . . . .15,16,19,24,27,. . . . . .
. . . . . . . . . . . .20,23,28,33,34,. . .
. . . . . . . . . . . .21,22,29,32,35,. . .
. . . . . . . . . . . . . . ,30,31,36,. . .
. . . . . . . . . . . . . . . . . .37,. . .

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A162200, A162201, A162202, A162203, A162340, A162341, A162342, A162343, A162345, A162350, A162351, A162352.

A162345 Length of n-th edge in the graph of the zig-zag function for prime numbers.

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 3, 3, 5, 6, 4, 4, 5, 3, 4, 5, 5, 7, 6, 3, 3, 3, 3, 9, 9, 5, 4, 6, 6, 4, 6, 5, 5, 6, 4, 6, 6, 3, 3, 7, 12, 8, 3, 3, 5, 4, 6, 8, 6, 6, 4, 4, 5, 3, 6, 12, 9, 3, 3, 9, 10, 8, 6, 3, 5, 7, 7, 6, 5, 5, 7, 6, 6, 9, 6, 6, 6, 4, 5, 5

Offset: 1

Omar E. Pol, Jul 04 2009

Also, first differences of A162800.
Also {2, 2, } together with the numbers A052288.
Note that the graph of the zig-zag function for prime numbers is similar to the graph of the mountain path function for prime numbers but with exactly a vertex between consecutive odd noncomposite numbers (A006005).
This is the same as A115061 if n>1 (and also essentially equal to A052288). Proof: Because this is the first differences of A162800, which is {0,2} together with A024675, this sequence (for n>=3) is given by a(n) = (prime(n+1) - prime(n-1))/2. Similarly, because half the numbers between prime(n-1) and prime(n+1) are closer to prime(n) than any other prime, A115061(n) = (prime(n+1) - prime(n-1))/2 for n>=3 as well. - Nathaniel Johnston, Jun 25 2011

			Array begins:
=====
x, y
=====
2, 2;
2, 3;
3, 3;
3, 3;
5, 4;

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Omar E. Pol, Graph of the mountain path function for prime numbers

Cf. A000040, A006005, A008578, A024675, A052288, A162203, A162800, A162801, A162802.

[2,2] cat[(NthPrime(n+1)-NthPrime(n-1))/2: n in [3..80]]; // Vincenzo Librandi, Dec 19 2016

A162345 := proc(n) if(n<=2)then return 2: fi: return (ithprime(n+1) - ithprime(n-1))/2: end: seq(A162345(n),n=1..100); # Nathaniel Johnston, Jun 25 2011

Join[{2, 2}, Table[(Prime[n+1] - Prime[n-1])/2, {n, 3, 100}]] (* Vincenzo Librandi, Dec 19 2016 *)

a(n) = (prime(n+1) - prime(n-1))/2 for n>=3. - Nathaniel Johnston, Jun 25 2011

Edited by Omar E. Pol, Jul 16 2009

A162350 The path of the primes: Pairs (x,y) such that the points P(x,y) represent the position of the nonnegative integers in the graph of the "mountain path" function for prime numbers.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Jul 03 2009

			n ..... Point
0 ..... P(0,0)
1 ..... P(1,0)
2 ..... P(2,0)
3 ..... P(2,1)
4 ..... P(2,2)
5 ..... P(3,2)
6 ..... P(4,2)
7 ..... P(4,3)
8 ..... P(4,4)
9 ..... P(4,5)
10 .... P(5,5)
11 .... P(5,4)

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The path of the primes

Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162203, A162340, A162341, A162342, A162343, A162344, A162351, A162352.

cp's OEIS Frontend

A162200 Number on the positive y axis of the n-th horizontal component in the graph of the "mountain path" function for prime numbers.

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A162201 First differences of A162200.

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A162202 Number of the n-th vertex in the graph of the "mountain path" function for prime numbers.

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A162340 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the graph of the "mountain path" function for prime numbers.

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A162341 a(n) = number of grid points P(x,y) that are covered by a polyedge as the graph of the "mountain path" function for prime numbers, where x=n and y=0..oo.

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A162342 Partial sums of A162200.

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A162343 Array read by rows in which row n lists the numbers that are in the same "y" level in the mountain path of the primes.

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A162344 Array read by rows in which row n lists the numbers that are in the same "x" level in the mountain path of the primes.

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A162345 Length of n-th edge in the graph of the zig-zag function for prime numbers.

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