cp's OEIS Frontend

A309663 Primes that begin a run of consecutive primes whose first differences are nondecreasing.

%I A309663 #54 Jul 29 2020 18:51:35
%S A309663 2,13,19,31,41,61,71,83,101,109,131,139,151,167,181,193,199,227,241,
%T A309663 257,271,281,311,337,349,373,383,401,421,433,443,461,487,503,523,547,
%U A309663 563,571,593,601,617,641,661,677,709,727,743,757,773,797,811,823,829,857
%N A309663 Primes that begin a run of consecutive primes whose first differences are nondecreasing.
%C A309663 Arrange primes in rows where the value of the increase between consecutive primes cannot shrink.
%C A309663 Conjecture: Average length of each run = e, converging from above.
%C A309663 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.
%H A309663 Reddit blog, <a href="https://www.reddit.com/r/math/comments/cbd0a7/found_e_in_the_primes_maybe/">Found "e" in the primes (maybe)</a>, 2019.
%e A309663 The first run is 2, 3, 5, 7, 11, thus the first value is 2;
%e A309663 The second run is 13, 17, thus the second value is 13;
%e A309663 The third run is 19, 23, 29, thus the third value is 19;
%e A309663 The fourth run is 31, 37, thus the fourth value is 31.
%o A309663 (PARI) lista(nn) = {my(d=m=2); forprime(p=2, nn, if(p-m<d, d=0; print1(p, ", "), d=p-m); m=p); } \\ _Jinyuan Wang_, Jul 09 2020
%Y A309663 Same process as A331544 except this allows equal increases in the runs.
%K A309663 nonn
%O A309663 1,1
%A A309663 _Peter Woodward_, Jun 06 2020
%E A309663 More terms from _Jinyuan Wang_, Jul 09 2020