

A309663


Primes that begin a run of consecutive primes whose first differences are nondecreasing.


1



2, 13, 19, 31, 41, 61, 71, 83, 101, 109, 131, 139, 151, 167, 181, 193, 199, 227, 241, 257, 271, 281, 311, 337, 349, 373, 383, 401, 421, 433, 443, 461, 487, 503, 523, 547, 563, 571, 593, 601, 617, 641, 661, 677, 709, 727, 743, 757, 773, 797, 811, 823, 829, 857
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OFFSET

1,1


COMMENTS

Arrange primes in rows where the value of the increase between consecutive primes cannot shrink.
Conjecture: Average length of each run = e, converging from above.
Heuristic justification: Consider that e = 2 + 1/2! + 1/3! + 1/4! + ... We always have at least two values in a run. The odds of there being a third value = 1/2!. The odds of there being a fourth value = 1/3! (because exactly 3! ways to sort three differences, and only one of these ways is in increasing order). etc... The process is aberrated by the possibility of equal increases. This sequence allows equal increases in the runs, causing the convergence to e to approach from above. However, as the scale increases, these equal increases occur less frequently and their effect approaches zero. The sister sequence, "Primes that begin a run of consecutive primes whose first differences are strictly increasing", disallows equal increases in its rows, thus it approaches e from below. Averaging the runs of the two sequences negates the aberration, giving immediate convergence to e.


LINKS

Table of n, a(n) for n=1..54.
Reddit blog, Found "e" in the primes (maybe), 2019.


EXAMPLE

The first run is 2, 3, 5, 7, 11, thus the first value is 2;
The second run is 13, 17, thus the second value is 13;
The third run is 19, 23, 29, thus the third value is 19;
The fourth run is 31, 37, thus the fourth value is 31.


PROG

(PARI) lista(nn) = {my(d=m=2); forprime(p=2, nn, if(pm<d, d=0; print1(p, ", "), d=pm); m=p); } \\ Jinyuan Wang, Jul 09 2020


CROSSREFS

Same process as A331544 except this allows equal increases in the runs.
Sequence in context: A298089 A067208 A142348 * A347259 A181686 A206462
Adjacent sequences: A309660 A309661 A309662 * A309664 A309665 A309666


KEYWORD

nonn


AUTHOR

Peter Woodward, Jun 06 2020


EXTENSIONS

More terms from Jinyuan Wang, Jul 09 2020


STATUS

approved



