A258039 Numbers prime(k) such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

2, 3, 5, 11, 17, 23, 31, 41, 47, 53, 61, 71, 79, 89, 101, 103, 109, 127, 137, 149, 157, 167, 173, 181, 193, 199, 227, 233, 241, 257, 269, 277, 283, 307, 313, 331, 347, 353, 367, 379, 389, 401, 419, 431, 439, 449, 461, 467, 487, 499, 509, 541, 557, 569, 577

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

Links

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

Crossrefs

Cf. A258036, A258037, A258038.

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(A258037(n)). - Jason Yuen, Nov 13 2024