A258038 Numbers prime(k) such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

7, 13, 19, 29, 37, 43, 59, 67, 73, 83, 97, 107, 113, 131, 139, 151, 163, 179, 191, 197, 211, 223, 229, 239, 251, 263, 271, 281, 293, 311, 317, 337, 349, 359, 373, 383, 397, 409, 421, 433, 443, 457, 463, 479, 491, 503, 521, 523, 547, 563, 571, 587, 599, 607

Offset: 1

Author

Clark Kimberling, Jun 05 2015

Keywords

• nonn
• easy

Comments

Partition of the positive integers: A258036, A258037;

Corresponding partition of the primes: A258038, A258039.

Examples

			D(prime(2), 1) = 3 - 2 > 0;
D(prime(3), 2) = 5 - 2*3 + 2 > 0;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = prime(4) = 7;
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A258036, A258037, A258039.

Programs

Mathematica

u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
p1 = Prime[w1]  (* A258038 *)
p2 = Prime[w2]  (* A258039 *)

Formula

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]

a(n) = prime(A258036(n)). - Jason Yuen, Nov 13 2024