A258037 Numbers k such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

1, 2, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 27, 29, 31, 33, 35, 37, 39, 40, 42, 44, 46, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121

Offset

1,2

Comments

Partition of the positive integers: A258036, A258037;

Corresponding partition of the primes: A258038, A258039.

Conjecture: all the terms of the difference sequence of A258037 belong to {1,2,3}.

Links

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

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]

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A258036, A258038, A258039.

Sequence in context: A066935 A042943 A306466 * A186330 A153809 A355330

Adjacent sequences: A258034 A258035 A258036 * A258038 A258039 A258040

Keyword

nonn,easy

Author

Clark Kimberling, Jun 05 2015

Status

approved