A337567 - OEIS

%I #20 Sep 05 2020 21:36:31

%S 1,2,3,5,7,10,13,17,22,28,35,43,52,61,71,82,94,107,121,136,151,167,

%T 184,202,221,241,262,283,305,328,352,377,403,430,457,485,514,544,575,

%U 607,640,673,707,742,778,815,853,892,932,973,1015,1058,1102,1147,1193,1240

%N Let a(0) = 1, k(0) = 1. For n >= 1; if a(n-1) + k(n-1) is a prime, then a(n) = a(n-1) + k(n-1), k(n) = k(n-1);  else a(n) = a(n-1) + k(n-1) + 1, k(n) = k(n-1) + 1.

%C Empirically a(n) ~ (e*n/4)^2; n-th gap a(n) - a(n-1) = k(n) ~ sqrt(e*n); the number of primes pi(a(n)) = n^(2*e/7).

%e a(0) = 1, k(0) = 1;

%e 1 + 1 = 2 is prime, a(1) = 2, k(1) = 1;

%e 2 + 1 = 3 is prime, a(2) = 3, k(2) = 1;

%e 3 + 1 = 4 is not a prime, a(3) = 3 + 1 + 1 = 5, k(3) = 2;

%e 5 + 2 = 7 is prime, a(4) = 7, k(4) = 2;

%e 7 + 2 = 9 is not a prime, a(5) = 7 + 2 + 1 = 10, k(5) = 3;

%e and so on.

%t a[0] = {1, 1}; a[n_] := a[n] = If[PrimeQ[(s = Plus @@ a[n - 1])], {s, a[n - 1][[2]]}, {s, a[n - 1][[2]]} + 1]; First /@ Array[a, 50, 0] (* _Amiram Eldar_, Sep 03 2020 *)

%o (PARI) lista(nn) = {my(a = 1, k = 1, list = List()); for (n=1, nn, listput(list, a); if (isprime(a+k), a += k, a += k+1; k++);); Vec(list);} \\ _Michel Marcus_, Sep 01 2020

%Y Cf. A000040.

%K nonn

%O 0,2

%A _Ctibor O. Zizka_, Sep 01 2020

%E More terms from _Michel Marcus_, Sep 01 2020