A258039 - OEIS

%I #8 Nov 14 2024 08:20:03

%S 2,3,5,11,17,23,31,41,47,53,61,71,79,89,101,103,109,127,137,149,157,

%T 167,173,181,193,199,227,233,241,257,269,277,283,307,313,331,347,353,

%U 367,379,389,401,419,431,439,449,461,467,487,499,509,541,557,569,577

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

%C Partition of the positive integers: A258036, A258037;

%C Corresponding partition of the primes: A258038, A258039.

%H Clark Kimberling, <a href="/A258039/b258039.txt">Table of n, a(n) for n = 1..1000</a>

%F 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]

%F a(n) = prime(A258037(n)). - _Jason Yuen_, Nov 13 2024

%e D(prime(2), 1) = 3 - 2 > 0, so a(1) = prime(1) = 2;

%e D(prime(3), 2) = 5 - 2*3 + 2 > 0, so a(2) = prime(2) = 3;

%e D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0.

%t u = Table[Prime[Range[k]], {k, 1, 1000}];

%t v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];

%t w1 = Flatten[Position[v, -1]] (* A258036 *)

%t w2 = Flatten[Position[v, 1]]  (* A258037 *)

%t p1 = Prime[w1]  (* A258038 *)

%t p2 = Prime[w2]  (* A258039 *)

%Y Cf. A258036, A258037, A258038.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 05 2015