A330545 a(1) = 2; thereafter a(n) = a(n-1) + (-1)^(n + 1)*(prime(n) - prime(n - 1) - 1) (where prime(k) denotes the k-th prime).

2, 2, 3, 2, 5, 4, 7, 6, 9, 4, 5, 0, 3, 2, 5, 0, 5, 4, 9, 6, 7, 2, 5, 0, 7, 4, 5, 2, 3, 0, 13, 10, 15, 14, 23, 22, 27, 22, 25, 20, 25, 24, 33, 32, 35, 34, 45, 34, 37, 36, 39, 34, 35, 26, 31, 26, 31, 30, 35, 32, 33, 24, 37, 34, 35, 32, 45, 40, 49, 48, 51, 46, 53, 48, 53, 50, 55, 48, 51, 44, 53

Offset: 1

Author

N. J. A. Sloane, Dec 17 2019

Keywords

• sign
• easy

Comments

a(n) is the column of the Boustrophedon triangle in A330339 that contains prime(n).

If a(n) = 0 then p = prime(n) is a term of A330339, and n is a term of A330546.

Since this has a simple recurrence, it is the key to understanding A330339. However, note that this sequence in turn can be simply expressed in terms of the classic sequence A008347:

a(n) = prime(n) + 1 - 2 * A008347(n) if n is even,

a(n) = 2 * A008347(n) - prime(n) if n is odd.

The sequence that ties all these sequences together is A330547 (q.v.).

First negative term is a(146) = -2.

Note on the links: The illustrations from Angelini and Trump show all the terms 0,1,2,3,4,... (as in A330339), while those of Havermann, Sloane, and Stevenson just show the primes.

The column number mod 4 uniquely determines the residue class of primes mod 4 in that column. If the column number is 0,1,2,3 mod 4 then the primes in that column are 1,3,3,1 respectively (see the "Notes" link). - N. J. A. Sloane, Jan 04 2020

For large n, the graphs of A330545 and A330547 are essentially identical.

Based on the first 10^12 terms, it appears that lim sup |a(n)| is about n^(2/3). This estimate is based on the plots below by Sloane, Trump (the first plot), Havermann (the first plot), and Stevenson (all three plots). - N. J. A. Sloane, Jan 21 2020

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000
• Eric Angelini, Illustration of beginning of the triangle in A330339.
• Hans Havermann, Plot of 4*10^8 terms of A330545, sampled every 1000 terms, points joined.
• Hans Havermann, More detailed view of terms of A330545 from 290 million to 310 million, sampled every 10 terms, points joined.
• N. J. A. Sloane, Illustration of first 16 rows of A330545.
• N. J. A. Sloane, Notes on the sequence of Bostrophedon primes (A330399) and the "ski-run" A330545.
• N. J. A. Sloane, State diagram for columns of A330545.
• Frank Stevenson, Plot of first 10^10 terms of A330547, sampled every 10^5 steps. Havermann's plot of 4*10^8 terms appears at the very start of this plot.
• Frank Stevenson, Plot of first 10^11 terms of A330547, sampled every 10^6 steps. The preceding plot of 10^10 terms can be seen to the left of the blue line.
• Frank Stevenson, Plot of first 10^12 terms of A330547, sampled every 10^7 steps. The preceding plot of 10^11 terms can be seen to the left of the blue line.
• Walter Trump, An extended picture of the triangle in A330339, showing the first 550 rows, down to the row that starts 3989. [The zeroth column is just to the right of the vertical red line. Note that after a while the rows extend to the left of the red line. The digits are too small to be read.]
• Walter Trump, An extended picture of the triangle in A330339, showing the first 550 rows, down to the row that starts 3989. [Same picture as the previous one, but with 6 red dots added to show the primes in column 0.]

Crossrefs

Cf. A000040, A008347, A014692, A330339, A330546, A330547.

Programs

Mathematica

A330545[1] := 2; A330545[n_] := A330545[n] = (A330545[n - 1] + (-1)^(n + 1) * (Prime[n] - Prime[n - 1] - 1)); Table[A330545[n], {n, 100}] (* Alonso del Arte, Dec 27 2019 *)

Formula

G.f.: G(-x)*(x+1)/(x-1), where G(x) = 2*x + 2*x^2 +3*x^3 + 4*x^4 + 7*x^5 + ... is the g.f. for A014692, {prime(n) - (n-1): n >= 1}.

Extensions

Keyword nonn changed to sign by Hans Havermann, Dec 27 2019