A162800 -id:A162800 - cp's OEIS frontend

A162801 Bisection of A162800.

0, 4, 9, 15, 21, 30, 39, 45, 56, 64, 72, 81, 93, 102, 108, 120, 134, 144, 154, 165, 176, 186, 195, 205, 225, 231, 240, 254, 266, 274, 282, 300, 312, 324, 342, 351, 363, 376, 386, 399, 414, 426, 436, 446, 459, 465, 483, 495, 506, 522, 544, 560, 570, 582, 596

Offset: 1

Omar E. Pol, Jul 16 2009

Essentially the same as A058296.

Cf. A058296, A079424, A162345, A162802.

A162802 Bisection of A162800.

Original entry on oeis.org

2, 6, 12, 18, 26, 34, 42, 50, 60, 69, 76, 86, 99, 105, 111, 129, 138, 150, 160, 170, 180, 192, 198, 217, 228, 236, 246, 260, 270, 279, 288, 309, 315, 334, 348, 356, 370, 381, 393, 405, 420, 432, 441, 453, 462, 473, 489, 501, 515, 532, 552, 566, 574, 590, 600

Offset: 1

Omar E. Pol, Jul 16 2009

Also, 2 together with the numbers A079424.

Cf. A079424, A162345, A162800, A162801.

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

A162735 An alternating sum of all numbers from prime(n) to prime(n+1).

Original entry on oeis.org

1, 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279

Offset: 1

Juri-Stepan Gerasimov, Jul 13 2009

nonn

Without the initial term, identical to A024675, cf. formula. Except for the initial terms, also the same as A162800. - M. F. Hasler, Jun 01 2013

			a(1) = 1 = -2+3. a(2) = 4=3-4+5. a(3) = 6 =5-6+7. a(4) = 9 = 7-8+9-10+11.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A024675, A162800.

Join[{1}, Most[#] + Differences[#]/2] & [Prime[Range[2, 100]]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = sum_{j= A000040(n).. A000040(n+1)} (-1)^(j+1)*j = A001057(A000040(n+1))-A001057(A000040(n)-1).

{1} U A024675.

Edited by R. J. Mathar, Sep 23 2009

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A162801 Bisection of A162800.

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A162802 Bisection of A162800.

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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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A162735 An alternating sum of all numbers from prime(n) to prime(n+1).

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