

A282178


Primes for which the sum of all preceding oddindexed prime gaps is exactly one greater than the sum of all preceding evenindexed prime gaps.


4



3, 7, 43, 79, 107, 1471, 1579, 1663, 3491, 3547, 3659, 3691, 3719, 3779, 3823, 3851, 3947, 4079, 4583, 4679, 4703, 27271, 28643, 28663, 28711, 29023, 41603, 41651, 41999, 42443, 42787, 42899, 44263, 44279, 45971, 50599, 133979, 28335623
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OFFSET

1,1


COMMENTS

If the counting numbers 1, 2, 3, ... are written out sequentially such that one unit is moved in a given direction each time a new number is written and such that the direction is reversed if and only if a prime number is reached, these are the primes that lie directly below the number 1.
Let p(k) = kth prime, Delta p(k) = p(k+1)p(k). The sequence contains those primes q such that
Sum_{k odd, p(k+1) <= q} Delta p(k) = 1 + Sum_{k even, p(k+1) <= q} Delta p(k).
The boustrophedon path described in the first comment can be drawn as follows (it is very similar to the path in A330339):
2.1 0..1..2..3..4..5..6..7..8..

......1..2
......3
.........4..5
......7..6
.........8..9.10.11
...........13.12
..............14.15.16.17
.................19.18
....................20.21.22.23
...........29.28.27.26.25.24
..............30.31
37.3635.34.33.32
...
The primes that fall in column 0 make up the sequence.
Thanks to Walter Trump for pointing out that this sequence is very similar to the Boustrophedon Primes sequence of A330339, and for correcting an omission in an earlier version of these comments.
The close relationship between the two sequences is demonstrated by the fact that the Boustrophedon Primes occur exactly when A330545 is 0, whereas the primes in the present sequence occur exactly when A330545 is 1 or 2.
Yet another way to relate the two sequences is to say that the present sequence gives all the primes > 2 in columns 1 and 2 of the triangle in A330339.
(End)
The primes (other than 2) occur only in evennumbered columns: primes congruent to 3 mod 4 occur in columns congruent to 0 mod 4, and primes congruent to 1 mod 4 occur in columns congruent to 2 mod 4. See the "Notes" link for proof. In particular, a(n) == 3 mod 4. N. J. A. Sloane, Jan 04 2020
Frank Stevenson's data seems to suggest that a(n) is roughly growing like n^c where c is about 2.74.  N. J. A. Sloane, Dec 31 2019


LINKS



MATHEMATICA

With[{s = Differences@ Prime@ Range[10^5]}, Prime[1 + Position[Array[Total@ Take[s, {1, #, 2}]  Total@ Take[s, {2, #, 2}] &, Length@ s], 1][[All, 1]] ] ]


PROG

(PARI) my(a=2, n=1, pp=2); forprime(p=3, 47000000, n++; a+= (1)^(n+1)*(ppp); if(a==1, print1(p, ", ")); pp=p) \\ Hugo Pfoertner, Dec 23 2019


CROSSREFS

The indices of these primes are given by A127596.


KEYWORD

nonn


AUTHOR



STATUS

approved



