A258036 Numbers k such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 101, 103, 105, 107, 109, 111, 114, 116, 118, 120, 122, 124, 126, 128

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.

Do all the terms of the difference sequence of A258036 belong to {1,2,3}?

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) = 4;
		

Links

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

Crossrefs

Cf. A258037, A258038, 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 *)
Prime[w1]  (* A258038 *)
Prime[w2]  (* A258039 *)

PARI

is(k) = {my(p=primes(k));sum(i=0,k-1,(-1)^i*p[k-i]*binomial(k-1,i))<0} \\ Jason Yuen, Nov 13 2024

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]