A282178 -id:A282178 - cp's OEIS frontend

A330554 Union of 2, A282178, and A330339.

2, 3, 7, 37, 43, 53, 79, 89, 107, 113, 1471, 1579, 1663, 3491, 3547, 3659, 3691, 3719, 3779, 3821, 3823, 3851, 3947, 3989, 4079, 4583, 4657, 4679, 4703, 27271, 28643, 28661, 28663, 28711, 29021, 29023, 41603, 41641, 41651, 41669, 41999, 42443, 42787, 42899, 44249, 44263, 44279, 45971, 50599, 50909, 56053, 57041

Offset: 1

N. J. A. Sloane, Dec 22 2019

nonn

Equivalently, primes that occur in columns 0, 1, or 2 of the triangle in A330339. [Corrected by Walter Trump, Dec 23 2019]

The asymptotic behavior of A282178 and A330339 is a mystery. It is not even known if they are infinite. They are closely related. A282178 contains primes == 3 mod 4, and A330339 primes == 1 mod 4. Perhaps by combining them in this way some properties will become more visible.

Giovanni Resta, Table of n, a(n) for n = 1..25000 (first 1269 terms from N. J. A. Sloane)

Cf. A282178, A330339, A330545, A330547.

A330339 Boustrophedon primes: write the numbers 0, 1, 2, 3, ... in a triangle on a square grid in the boustrophedon manner, ending a row when a prime is reached; sequence lists primes that appear in the zeroth column.

Original entry on oeis.org

37, 53, 89, 113, 3821, 3989, 4657, 28661, 29021, 41641, 41669, 44249, 50909, 56053, 57041, 57301, 133981, 16501361, 46178761, 47633441, 47633477, 47722049, 47736121, 47774621, 47803477, 47810209, 47835013, 47835341, 47854969, 47862413, 47865017, 49448573, 49448617

Offset: 1

N. J. A. Sloane, Dec 17 2019, following a suggestion from Eric Angelini. a(5) and a(6) were found by Walter Trump. a(7)-a(17) from N. J. A. Sloane, Dec 17 2019

nonn

Eric Angelini's illustration shows the first 19 rows of the triangle. Each row ends when a prime is reached, and the next row starts directly under this prime but moves in the opposite direction.
The extended illustration from Walter Trump resembles a giant ski run.
Hans Havermann's plots of A330545, linked here, extend Walter Trump's graph to 4*10^8 rows (probably the longest ski run in the world). Only the turns are shown, and the illustration has been turned sideways.
A330545(k) = 0 iff prime(k) is a term of the present sequence. In a sense A330545 and the simpler A330547 are the more fundamental sequences and show the connection between the present problem and the ordinary primes and their alternating sums.
Note that because primes > 2 are odd, a prime can only appear in column 0 at the end of a row that is moving towards the left. A prime appearing in a row moving to the right will always appear in an odd-numbered column (and in particular, not in the zero column).
Furthermore 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). In particular, a(n) == 1 mod 4. - N. J. A. Sloane, Jan 04 2020
Note that the primes > 2 in column one and two are the primes in A282178.
Note on the links: The illustrations from Angelini and Trump show all the terms 0,1,2,3,4,..., while those of Havermann and Sloane just show the primes (as in A330545).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 516 terms from Hans Havermann)
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 (A330339) and the "ski-run" A330545.
N. J. A. Sloane, State diagram for columns of A330545.
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.]

Cf. A282178, A330545, A330547.
A330546 gives the list of indices i such that a(n) = prime(i).
A127596 is another sequence with a similar flavor.
Not to be confused with A000747 = Boustrophedon transform of primes.

More terms from Hans Havermann, Dec 17 2019

A127596 Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.

Original entry on oeis.org

2, 4, 14, 22, 28, 233, 249, 261, 488, 497, 511, 515, 519, 526, 531, 534, 548, 562, 620, 633, 635, 2985, 3119, 3123, 3128, 3157, 4350, 4358, 4392, 4438, 4474, 4484, 4606, 4610, 4759, 5191, 12493, 1761067, 2785124, 2785152, 2785718, 2785729, 2867471

Offset: 1

Manuel Valdivia, Apr 03 2007

nonn

Or, with prime(0) = 1, numbers k such that Sum_{i=0..k-1} (prime(i+1)-prime(i))*(-1)^i = Sum_{i=0..k-1} (A008578(i+1)-A008578(i))*(-1)^i = 0.

There are 313 terms < 10^7, 846 terms < 10^8, 1161 terms < 10^9.

			1 - A001223(1) = 1 - 1 = 0, hence 2 is a term.
1 - A001223(1) + A001223(2) - A001223(3) = 1 - 1 + 2 - 2 = 0, hence 4 is a term.

Jinyuan Wang, Table of n, a(n) for n = 1..1161 (first 846 terms from Klaus Brockhaus)
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Prime Difference Function

Cf. A001223 (differences between consecutive primes), A008578 (prime numbers at the beginning of the 20th century), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end).

Cf. A282178 (prime(a(n))), A330545, A330547.

S=0; Do[j=Prime[n+1]; i=Prime[n]; d[n]=j-i; S=S+(d[n]*(-1)^n); If[S+1==0, Print[Table[j|PrimePi[j]|S+1]]], {n,1,10^7,1}]

{m=10^8; n=1; p=1; e=1; s=0; while(nKlaus Brockhaus, Apr 29 2007 */

Edited by Klaus Brockhaus, Apr 29 2007

cp's OEIS Frontend

A330554 Union of 2, A282178, and A330339.

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A330339 Boustrophedon primes: write the numbers 0, 1, 2, 3, ... in a triangle on a square grid in the boustrophedon manner, ending a row when a prime is reached; sequence lists primes that appear in the zeroth column.

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A127596 Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.

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